Calculus Examples

$g (t) = \frac{{(3 + 6 e^{8 t})}^{14}}{\sqrt[3]{t^{7} - \ln ({(t)}^{52})}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{t^{7} - \ln (t^{52})}$ as ${(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}$ .

$\frac{d}{d t} [\frac{{(3 + 6 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = {(3 + 6 e^{8 t})}^{14}$ and $g (t) = {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}$ .

$\frac{{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} \frac{d}{d t} [{(3 + 6 e^{8 t})}^{14}] - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{({(t^{7} - \ln (t^{52}))}^{\frac{1}{3}})}^{2}}$

Step 3

Multiply the exponents in ${({(t^{7} - \ln (t^{52}))}^{\frac{1}{3}})}^{2}$ .

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Step 3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} \frac{d}{d t} [{(3 + 6 e^{8 t})}^{14}] - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{1}{3} \cdot 2}}$

Step 3.2

Combine $\frac{1}{3}$ and $2$ .

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{14}$ and $g (t) = 3 + 6 e^{8 t}$ .

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Step 4.1

To apply the Chain Rule, set $u_{1}$ as $3 + 6 e^{8 t}$ .

$\frac{{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} (\frac{d}{d u_{1}} [{u_{1}}^{14}] \frac{d}{d t} [3 + 6 e^{8 t}]) - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 14$ .

$\frac{{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} (14 {u_{1}}^{13} \frac{d}{d t} [3 + 6 e^{8 t}]) - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 4.3

Replace all occurrences of $u_{1}$ with $3 + 6 e^{8 t}$ .

$\frac{{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} (14 {(3 + 6 e^{8 t})}^{13} \frac{d}{d t} [3 + 6 e^{8 t}]) - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 5

Differentiate.

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Step 5.1

Move $14$ to the left of ${(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}$ .

$\frac{14 \cdot {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} \frac{d}{d t} [3 + 6 e^{8 t}] - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 5.2

By the Sum Rule, the derivative of $3 + 6 e^{8 t}$ with respect to $t$ is $\frac{d}{d t} [3] + \frac{d}{d t} [6 e^{8 t}]$ .

$\frac{14 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} (\frac{d}{d t} [3] + \frac{d}{d t} [6 e^{8 t}]) - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 5.3

Since $3$ is constant with respect to $t$ , the derivative of $3$ with respect to $t$ is $0$ .

$\frac{14 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} (0 + \frac{d}{d t} [6 e^{8 t}]) - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 5.4

Add $0$ and $\frac{d}{d t} [6 e^{8 t}]$ .

$\frac{14 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} \frac{d}{d t} [6 e^{8 t}] - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 5.5

Since $6$ is constant with respect to $t$ , the derivative of $6 e^{8 t}$ with respect to $t$ is $6 \frac{d}{d t} [e^{8 t}]$ .

$\frac{14 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} (6 \frac{d}{d t} [e^{8 t}]) - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 5.6

Multiply $6$ by $14$ .

$\frac{84 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} \frac{d}{d t} [e^{8 t}] - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 6

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = e^{t}$ and $g (t) = 8 t$ .

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Step 6.1

To apply the Chain Rule, set $u_{2}$ as $8 t$ .

$\frac{84 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d t} [8 t]) - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 6.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$\frac{84 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} (e^{u_{2}} \frac{d}{d t} [8 t]) - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 6.3

Replace all occurrences of $u_{2}$ with $8 t$ .

$\frac{84 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} (e^{8 t} \frac{d}{d t} [8 t]) - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 7

Differentiate.

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Step 7.1

Since $8$ is constant with respect to $t$ , the derivative of $8 t$ with respect to $t$ is $8 \frac{d}{d t} [t]$ .

$\frac{84 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} (e^{8 t} (8 \frac{d}{d t} [t])) - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 7.2

Multiply $8$ by $84$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} (e^{8 t} (\frac{d}{d t} [t])) - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 7.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} (e^{8 t} \cdot 1) - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 7.4

Multiply $e^{8 t}$ by $1$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - {(3 + 6 e^{8 t})}^{14} \frac{d}{d t} [{(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 8

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{\frac{1}{3}}$ and $g (t) = t^{7} - \ln (t^{52})$ .

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Step 8.1

To apply the Chain Rule, set $u_{3}$ as $t^{7} - \ln (t^{52})$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - {(3 + 6 e^{8 t})}^{14} (\frac{d}{d u_{3}} [{u_{3}}^{\frac{1}{3}}] \frac{d}{d t} [t^{7} - \ln (t^{52})])}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 8.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = \frac{1}{3}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - {(3 + 6 e^{8 t})}^{14} (\frac{1}{3} {u_{3}}^{\frac{1}{3} - 1} \frac{d}{d t} [t^{7} - \ln (t^{52})])}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 8.3

Replace all occurrences of $u_{3}$ with $t^{7} - \ln (t^{52})$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - {(3 + 6 e^{8 t})}^{14} (\frac{1}{3} {(t^{7} - \ln (t^{52}))}^{\frac{1}{3} - 1} \frac{d}{d t} [t^{7} - \ln (t^{52})])}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - {(3 + 6 e^{8 t})}^{14} (\frac{1}{3} {(t^{7} - \ln (t^{52}))}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d t} [t^{7} - \ln (t^{52})])}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 10

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - {(3 + 6 e^{8 t})}^{14} (\frac{1}{3} {(t^{7} - \ln (t^{52}))}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d t} [t^{7} - \ln (t^{52})])}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 11

Combine the numerators over the common denominator.

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - {(3 + 6 e^{8 t})}^{14} (\frac{1}{3} {(t^{7} - \ln (t^{52}))}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d t} [t^{7} - \ln (t^{52})])}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 12

Simplify the numerator.

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Step 12.1

Multiply $- 1$ by $3$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - {(3 + 6 e^{8 t})}^{14} (\frac{1}{3} {(t^{7} - \ln (t^{52}))}^{\frac{1 - 3}{3}} \frac{d}{d t} [t^{7} - \ln (t^{52})])}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 12.2

Subtract $3$ from $1$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - {(3 + 6 e^{8 t})}^{14} (\frac{1}{3} {(t^{7} - \ln (t^{52}))}^{\frac{- 2}{3}} \frac{d}{d t} [t^{7} - \ln (t^{52})])}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 13

Differentiate.

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Step 13.1

Move the negative in front of the fraction.

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - {(3 + 6 e^{8 t})}^{14} (\frac{1}{3} {(t^{7} - \ln (t^{52}))}^{- \frac{2}{3}} \frac{d}{d t} [t^{7} - \ln (t^{52})])}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 13.2

Combine fractions.

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Step 13.2.1

Combine $\frac{1}{3}$ and ${(t^{7} - \ln (t^{52}))}^{- \frac{2}{3}}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - {(3 + 6 e^{8 t})}^{14} (\frac{{(t^{7} - \ln (t^{52}))}^{- \frac{2}{3}}}{3} \frac{d}{d t} [t^{7} - \ln (t^{52})])}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 13.2.2

Move ${(t^{7} - \ln (t^{52}))}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - {(3 + 6 e^{8 t})}^{14} (\frac{1}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} \frac{d}{d t} [t^{7} - \ln (t^{52})])}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 13.2.3

Combine $\frac{1}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$ and ${(3 + 6 e^{8 t})}^{14}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 + 6 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} \frac{d}{d t} [t^{7} - \ln (t^{52})]}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 13.3

By the Sum Rule, the derivative of $t^{7} - \ln (t^{52})$ with respect to $t$ is $\frac{d}{d t} [t^{7}] + \frac{d}{d t} [- \ln (t^{52})]$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 + 6 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (\frac{d}{d t} [t^{7}] + \frac{d}{d t} [- \ln (t^{52})])}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 13.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 7$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 + 6 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} + \frac{d}{d t} [- \ln (t^{52})])}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 13.5

Since $- 1$ is constant with respect to $t$ , the derivative of $- \ln (t^{52})$ with respect to $t$ is $- \frac{d}{d t} [\ln (t^{52})]$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 + 6 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} - \frac{d}{d t} [\ln (t^{52})])}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 14

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \ln (t)$ and $g (t) = t^{52}$ .

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Step 14.1

To apply the Chain Rule, set $u_{4}$ as $t^{52}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 + 6 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} - (\frac{d}{d u_{4}} [\ln (u_{4})] \frac{d}{d t} [t^{52}]))}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 14.2

The derivative of $\ln (u_{4})$ with respect to $u_{4}$ is $\frac{1}{u_{4}}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 + 6 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} - (\frac{1}{u_{4}} \frac{d}{d t} [t^{52}]))}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 14.3

Replace all occurrences of $u_{4}$ with $t^{52}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 + 6 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} - (\frac{1}{t^{52}} \frac{d}{d t} [t^{52}]))}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 15

Differentiate using the Power Rule.

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Step 15.1

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 52$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 + 6 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} - \frac{1}{t^{52}} (52 t^{51}))}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 15.2

Simplify terms.

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Step 15.2.1

Multiply $52$ by $- 1$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 + 6 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} - 52 \frac{1}{t^{52}} t^{51})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 15.2.2

Combine $- 52$ and $\frac{1}{t^{52}}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 + 6 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} + \frac{- 52}{t^{52}} t^{51})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 15.2.3

Combine $\frac{- 52}{t^{52}}$ and $t^{51}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 + 6 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} + \frac{- 52 t^{51}}{t^{52}})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 15.2.4

Cancel the common factor of $t^{51}$ and $t^{52}$ .

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Step 15.2.4.1

Factor $t^{51}$ out of $- 52 t^{51}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 + 6 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} + \frac{t^{51} \cdot - 52}{t^{52}})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 15.2.4.2

Cancel the common factors.

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Step 15.2.4.2.1

Factor $t^{51}$ out of $t^{52}$ .

Step 15.2.4.2.2

Cancel the common factor.

Step 15.2.4.2.3

Rewrite the expression.

Step 15.2.5

Move the negative in front of the fraction.

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 + 6 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} - \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16

Simplify.

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Step 16.1

Simplify the numerator.

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Step 16.1.1

Simplify the numerator.

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Step 16.1.1.1

Factor $3$ out of $3 + 6 e^{8 t}$ .

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Step 16.1.1.1.1

Factor $3$ out of $3$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 (1) + 6 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} - \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.1.1.2

Factor $3$ out of $6 e^{8 t}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 (1) + 3 (2 e^{8 t}))}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} - \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.1.1.3

Factor $3$ out of $3 (1) + 3 (2 e^{8 t})$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{{(3 (1 + 2 e^{8 t}))}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} - \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.1.2

Apply the product rule to $3 (1 + 2 e^{8 t})$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{3^{14} {(1 + 2 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} - \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.1.3

Raise $3$ to the power of $14$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{4782969 {(1 + 2 e^{8 t})}^{14}}{3 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} - \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.2

Expand $\ln (t^{52})$ by moving $52$ outside the logarithm.

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{4782969 {(1 + 2 e^{8 t})}^{14}}{3 {(t^{7} - (52 \ln (t)))}^{\frac{2}{3}}} (7 t^{6} - \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.3

Factor $3$ out of $4782969 {(1 + 2 e^{8 t})}^{14}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{3 (1594323 {(1 + 2 e^{8 t})}^{14})}{3 {(t^{7} - (52 \ln (t)))}^{\frac{2}{3}}} (7 t^{6} - \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.4

Cancel the common factors.

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Step 16.1.4.1

Factor $3$ out of $3 {(t^{7} - (52 \ln (t)))}^{\frac{2}{3}}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{3 (1594323 {(1 + 2 e^{8 t})}^{14})}{3 ({(t^{7} - (52 \ln (t)))}^{\frac{2}{3}})} (7 t^{6} - \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.4.2

Cancel the common factor.

Step 16.1.4.3

Rewrite the expression.

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - (52 \ln (t)))}^{\frac{2}{3}}} (7 t^{6} - \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.5

Simplify each term.

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Step 16.1.5.1

Multiply $- 1$ by $52$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - 52 \ln (t))}^{\frac{2}{3}}} (7 t^{6} - \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.5.2

Simplify $- 52 \ln (t)$ by moving $52$ inside the logarithm.

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6} - \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.6

Apply the distributive property.

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6}) - \frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (- \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.7

Multiply $- \frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (7 t^{6})$ .

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Step 16.1.7.1

Multiply $7$ by $- 1$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - 7 \frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} t^{6} - \frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (- \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.7.2

Combine $- 7$ and $\frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} + \frac{- 7 (1594323 {(1 + 2 e^{8 t})}^{14})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} t^{6} - \frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (- \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.7.3

Multiply $1594323$ by $- 7$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} + \frac{- 11160261 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} t^{6} - \frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (- \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.7.4

Combine $\frac{- 11160261 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$ and $t^{6}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} + \frac{- 11160261 {(1 + 2 e^{8 t})}^{14} t^{6}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} - \frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (- \frac{52}{t})}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.8

Multiply $- \frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} (- \frac{52}{t})$ .

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Step 16.1.8.1

Multiply $- 1$ by $- 1$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} + \frac{- 11160261 {(1 + 2 e^{8 t})}^{14} t^{6}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} + 1 \frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} \frac{52}{t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.8.2

Multiply $\frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$ by $1$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} + \frac{- 11160261 {(1 + 2 e^{8 t})}^{14} t^{6}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} + \frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} \cdot \frac{52}{t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.8.3

Multiply $\frac{1594323 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$ by $\frac{52}{t}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} + \frac{- 11160261 {(1 + 2 e^{8 t})}^{14} t^{6}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} + \frac{1594323 {(1 + 2 e^{8 t})}^{14} \cdot 52}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.8.4

Multiply $52$ by $1594323$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} + \frac{- 11160261 {(1 + 2 e^{8 t})}^{14} t^{6}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} + \frac{82904796 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.9

Move the negative in front of the fraction.

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{11160261 {(1 + 2 e^{8 t})}^{14} t^{6}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} + \frac{82904796 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.10

To write $- \frac{11160261 {(1 + 2 e^{8 t})}^{14} t^{6}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$ as a fraction with a common denominator, multiply by $\frac{t}{t}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{11160261 {(1 + 2 e^{8 t})}^{14} t^{6}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} \cdot \frac{t}{t} + \frac{82904796 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.11

Multiply $\frac{11160261 {(1 + 2 e^{8 t})}^{14} t^{6}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$ by $\frac{t}{t}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \frac{11160261 {(1 + 2 e^{8 t})}^{14} t^{6} t}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t} + \frac{82904796 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.12

Combine the numerators over the common denominator.

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} + \frac{- 11160261 {(1 + 2 e^{8 t})}^{14} t^{6} t + 82904796 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.13

Simplify the numerator.

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Step 16.1.13.1

Factor $1594323 {(1 + 2 e^{8 t})}^{14}$ out of $- 11160261 {(1 + 2 e^{8 t})}^{14} t^{6} t + 82904796 {(1 + 2 e^{8 t})}^{14}$ .

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Step 16.1.13.1.1

Factor $1594323 {(1 + 2 e^{8 t})}^{14}$ out of $- 11160261 {(1 + 2 e^{8 t})}^{14} t^{6} t$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} + \frac{1594323 {(1 + 2 e^{8 t})}^{14} (- 7 t^{6} t) + 82904796 {(1 + 2 e^{8 t})}^{14}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.13.1.2

Factor $1594323 {(1 + 2 e^{8 t})}^{14}$ out of $82904796 {(1 + 2 e^{8 t})}^{14}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} + \frac{1594323 {(1 + 2 e^{8 t})}^{14} (- 7 t^{6} t) + 1594323 {(1 + 2 e^{8 t})}^{14} (52)}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.13.1.3

Factor $1594323 {(1 + 2 e^{8 t})}^{14}$ out of $1594323 {(1 + 2 e^{8 t})}^{14} (- 7 t^{6} t) + 1594323 {(1 + 2 e^{8 t})}^{14} (52)$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} + \frac{1594323 {(1 + 2 e^{8 t})}^{14} (- 7 t^{6} t + 52)}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.13.2

Combine exponents.

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Step 16.1.13.2.1

Raise $t$ to the power of $1$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} + \frac{1594323 {(1 + 2 e^{8 t})}^{14} (- 7 (t^{1} t^{6}) + 52)}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.13.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} + \frac{1594323 {(1 + 2 e^{8 t})}^{14} (- 7 t^{1 + 6} + 52)}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.13.2.3

Add $1$ and $6$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} + \frac{1594323 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52)}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.14

To write $672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t}$ as a fraction with a common denominator, multiply by $\frac{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$ .

$\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} \cdot \frac{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} + \frac{1594323 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52)}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.15

Write each expression with a common denominator of $t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.1.15.1

Combine $672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t}$ and $\frac{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$ .

$\frac{\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} (t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} + \frac{1594323 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52)}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.15.2

Reorder the factors of ${(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} t$ .

$\frac{\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} (t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} + \frac{1594323 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52)}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.16

Combine the numerators over the common denominator.

$\frac{\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} (t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}) + 1594323 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52)}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17

Rewrite $\frac{672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} (t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}) + 1594323 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52)}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$ in a factored form.

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Step 16.1.17.1

Factor $3$ out of $672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} (t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}) + 1594323 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52)$ .

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Step 16.1.17.1.1

Factor $3$ out of $672 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} (t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}})$ .

$\frac{\frac{3 (224 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} (t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}})) + 1594323 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52)}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.1.2

Factor $3$ out of $1594323 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52)$ .

$\frac{\frac{3 (224 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} (t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}})) + 3 (531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.1.3

Factor $3$ out of $3 (224 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} (t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}})) + 3 (531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))$ .

$\frac{\frac{3 (224 {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} (t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}) + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.2

Multiply ${(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}$ by ${(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}$ by adding the exponents.

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Step 16.1.17.2.1

Move ${(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}$ .

$\frac{\frac{3 (224 ({(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} {(t^{7} - \ln (t^{52}))}^{\frac{1}{3}}) {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{3 (224 {(t^{7} - \ln (t^{52}))}^{\frac{2}{3} + \frac{1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.2.3

Combine the numerators over the common denominator.

$\frac{\frac{3 (224 {(t^{7} - \ln (t^{52}))}^{\frac{2 + 1}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.2.4

Add $2$ and $1$ .

$\frac{\frac{3 (224 {(t^{7} - \ln (t^{52}))}^{\frac{3}{3}} {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.2.5

Divide $3$ by $3$ .

$\frac{\frac{3 (224 {(t^{7} - \ln (t^{52}))}^{1} {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.3

Simplify $224 {(t^{7} - \ln (t^{52}))}^{1} {(3 + 6 e^{8 t})}^{13} e^{8 t} t$ .

$\frac{\frac{3 (224 (t^{7} - \ln (t^{52})) {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.4

Apply the distributive property.

$\frac{\frac{3 ((224 t^{7} + 224 (- \ln (t^{52}))) {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.5

Multiply $224 (- \ln (t^{52}))$ .

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Step 16.1.17.5.1

Multiply $- 1$ by $224$ .

$\frac{\frac{3 ((224 t^{7} - 224 \ln (t^{52})) {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.5.2

Simplify $- 224 \ln (t^{52})$ by moving $224$ inside the logarithm.

$\frac{\frac{3 ((224 t^{7} - \ln ({(t^{52})}^{224})) {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.6

Multiply the exponents in ${(t^{52})}^{224}$ .

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Step 16.1.17.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{3 ((224 t^{7} - \ln (t^{52 \cdot 224})) {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.6.2

Multiply $52$ by $224$ .

$\frac{\frac{3 ((224 t^{7} - \ln (t^{11648})) {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.7

Apply the distributive property.

$\frac{\frac{3 ((224 t^{7} {(3 + 6 e^{8 t})}^{13} - \ln (t^{11648}) {(3 + 6 e^{8 t})}^{13}) e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.8

Apply the distributive property.

$\frac{\frac{3 ((224 t^{7} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \ln (t^{11648}) {(3 + 6 e^{8 t})}^{13} e^{8 t}) t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.9

Apply the distributive property.

$\frac{\frac{3 (224 t^{7} {(3 + 6 e^{8 t})}^{13} e^{8 t} t - \ln (t^{11648}) {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.10

Multiply $t^{7}$ by $t$ by adding the exponents.

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Step 16.1.17.10.1

Move $t$ .

$\frac{\frac{3 (224 (t \cdot t^{7}) {(3 + 6 e^{8 t})}^{13} e^{8 t} - \ln (t^{11648}) {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.10.2

Multiply $t$ by $t^{7}$ .

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Step 16.1.17.10.2.1

Raise $t$ to the power of $1$ .

$\frac{\frac{3 (224 (t^{1} t^{7}) {(3 + 6 e^{8 t})}^{13} e^{8 t} - \ln (t^{11648}) {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{3 (224 t^{1 + 7} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \ln (t^{11648}) {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.10.3

Add $1$ and $7$ .

$\frac{\frac{3 (224 t^{8} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \ln (t^{11648}) {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.11

Apply the distributive property.

$\frac{\frac{3 (224 t^{8} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \ln (t^{11648}) {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7}) + 531441 {(1 + 2 e^{8 t})}^{14} \cdot 52)}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.12

Rewrite using the commutative property of multiplication.

$\frac{\frac{3 (224 t^{8} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \ln (t^{11648}) {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 \cdot - 7 {(1 + 2 e^{8 t})}^{14} t^{7} + 531441 {(1 + 2 e^{8 t})}^{14} \cdot 52)}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.13

Multiply $52$ by $531441$ .

$\frac{\frac{3 (224 t^{8} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \ln (t^{11648}) {(3 + 6 e^{8 t})}^{13} e^{8 t} t + 531441 \cdot - 7 {(1 + 2 e^{8 t})}^{14} t^{7} + 27634932 {(1 + 2 e^{8 t})}^{14})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.14

Multiply $531441$ by $- 7$ .

$\frac{\frac{3 (224 t^{8} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \ln (t^{11648}) {(3 + 6 e^{8 t})}^{13} e^{8 t} t - 3720087 {(1 + 2 e^{8 t})}^{14} t^{7} + 27634932 {(1 + 2 e^{8 t})}^{14})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15

Rewrite $224 t^{8} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \ln (t^{11648}) {(3 + 6 e^{8 t})}^{13} e^{8 t} t - 3720087 {(1 + 2 e^{8 t})}^{14} t^{7} + 27634932 {(1 + 2 e^{8 t})}^{14}$ in a factored form.

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Step 16.1.17.15.1

Factor $t {(3 + 6 e^{8 t})}^{13} e^{8 t}$ out of $224 t^{8} {(3 + 6 e^{8 t})}^{13} e^{8 t} - \ln (t^{11648}) {(3 + 6 e^{8 t})}^{13} e^{8 t} t$ .

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Step 16.1.17.15.1.1

Factor $t {(3 + 6 e^{8 t})}^{13} e^{8 t}$ out of $224 t^{8} {(3 + 6 e^{8 t})}^{13} e^{8 t}$ .

$\frac{\frac{3 (t {(3 + 6 e^{8 t})}^{13} e^{8 t} (224 t^{7}) - \ln (t^{11648}) {(3 + 6 e^{8 t})}^{13} e^{8 t} t - 3720087 {(1 + 2 e^{8 t})}^{14} t^{7} + 27634932 {(1 + 2 e^{8 t})}^{14})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.1.2

Factor $t {(3 + 6 e^{8 t})}^{13} e^{8 t}$ out of $- \ln (t^{11648}) {(3 + 6 e^{8 t})}^{13} e^{8 t} t$ .

$\frac{\frac{3 (t {(3 + 6 e^{8 t})}^{13} e^{8 t} (224 t^{7}) + t {(3 + 6 e^{8 t})}^{13} e^{8 t} (- \ln (t^{11648})) - 3720087 {(1 + 2 e^{8 t})}^{14} t^{7} + 27634932 {(1 + 2 e^{8 t})}^{14})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.1.3

Factor $t {(3 + 6 e^{8 t})}^{13} e^{8 t}$ out of $t {(3 + 6 e^{8 t})}^{13} e^{8 t} (224 t^{7}) + t {(3 + 6 e^{8 t})}^{13} e^{8 t} (- \ln (t^{11648}))$ .

$\frac{\frac{3 (t {(3 + 6 e^{8 t})}^{13} e^{8 t} (224 t^{7} - \ln (t^{11648})) - 3720087 {(1 + 2 e^{8 t})}^{14} t^{7} + 27634932 {(1 + 2 e^{8 t})}^{14})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.2

Factor $3$ out of $3 + 6 e^{8 t}$ .

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Step 16.1.17.15.2.1

Factor $3$ out of $3$ .

$\frac{\frac{3 (t {(3 (1) + 6 e^{8 t})}^{13} e^{8 t} (224 t^{7} - \ln (t^{11648})) - 3720087 {(1 + 2 e^{8 t})}^{14} t^{7} + 27634932 {(1 + 2 e^{8 t})}^{14})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.2.2

Factor $3$ out of $6 e^{8 t}$ .

$\frac{\frac{3 (t {(3 (1) + 3 (2 e^{8 t}))}^{13} e^{8 t} (224 t^{7} - \ln (t^{11648})) - 3720087 {(1 + 2 e^{8 t})}^{14} t^{7} + 27634932 {(1 + 2 e^{8 t})}^{14})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.2.3

Factor $3$ out of $3 (1) + 3 (2 e^{8 t})$ .

$\frac{\frac{3 (t {(3 (1 + 2 e^{8 t}))}^{13} e^{8 t} (224 t^{7} - \ln (t^{11648})) - 3720087 {(1 + 2 e^{8 t})}^{14} t^{7} + 27634932 {(1 + 2 e^{8 t})}^{14})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.3

Apply the product rule to $3 (1 + 2 e^{8 t})$ .

$\frac{\frac{3 (t \cdot 3^{13} {(1 + 2 e^{8 t})}^{13} e^{8 t} (224 t^{7} - \ln (t^{11648})) - 3720087 {(1 + 2 e^{8 t})}^{14} t^{7} + 27634932 {(1 + 2 e^{8 t})}^{14})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.4

Factor $531441 {(1 + 2 e^{8 t})}^{14}$ out of $- 3720087 {(1 + 2 e^{8 t})}^{14} t^{7} + 27634932 {(1 + 2 e^{8 t})}^{14}$ .

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Step 16.1.17.15.4.1

Factor $531441 {(1 + 2 e^{8 t})}^{14}$ out of $- 3720087 {(1 + 2 e^{8 t})}^{14} t^{7}$ .

$\frac{\frac{3 (t \cdot 3^{13} {(1 + 2 e^{8 t})}^{13} e^{8 t} (224 t^{7} - \ln (t^{11648})) + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7}) + 27634932 {(1 + 2 e^{8 t})}^{14})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.4.2

Factor $531441 {(1 + 2 e^{8 t})}^{14}$ out of $27634932 {(1 + 2 e^{8 t})}^{14}$ .

$\frac{\frac{3 (t \cdot 3^{13} {(1 + 2 e^{8 t})}^{13} e^{8 t} (224 t^{7} - \ln (t^{11648})) + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7}) + 531441 {(1 + 2 e^{8 t})}^{14} (52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.4.3

Factor $531441 {(1 + 2 e^{8 t})}^{14}$ out of $531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7}) + 531441 {(1 + 2 e^{8 t})}^{14} (52)$ .

$\frac{\frac{3 (t \cdot 3^{13} {(1 + 2 e^{8 t})}^{13} e^{8 t} (224 t^{7} - \ln (t^{11648})) + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.5

Factor ${(1 + 2 e^{8 t})}^{13}$ out of $t \cdot 3^{13} {(1 + 2 e^{8 t})}^{13} e^{8 t} (224 t^{7} - \ln (t^{11648})) + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52)$ .

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Step 16.1.17.15.5.1

Factor ${(1 + 2 e^{8 t})}^{13}$ out of $t \cdot 3^{13} {(1 + 2 e^{8 t})}^{13} e^{8 t} (224 t^{7} - \ln (t^{11648}))$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (t \cdot 3^{13} e^{8 t} (224 t^{7} - \ln (t^{11648}))) + 531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.5.2

Factor ${(1 + 2 e^{8 t})}^{13}$ out of $531441 {(1 + 2 e^{8 t})}^{14} (- 7 t^{7} + 52)$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (t \cdot 3^{13} e^{8 t} (224 t^{7} - \ln (t^{11648}))) + {(1 + 2 e^{8 t})}^{13} (531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.5.3

Factor ${(1 + 2 e^{8 t})}^{13}$ out of ${(1 + 2 e^{8 t})}^{13} (t \cdot 3^{13} e^{8 t} (224 t^{7} - \ln (t^{11648}))) + {(1 + 2 e^{8 t})}^{13} (531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52))$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (t \cdot 3^{13} e^{8 t} (224 t^{7} - \ln (t^{11648})) + 531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.6

Raise $3$ to the power of $13$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (t \cdot 1594323 e^{8 t} (224 t^{7} - \ln (t^{11648})) + 531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.7

Move $1594323$ to the left of $t$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (1594323 \cdot t e^{8 t} (224 t^{7} - \ln (t^{11648})) + 531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.8

Apply the distributive property.

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (1594323 t e^{8 t} (224 t^{7}) + 1594323 t e^{8 t} (- \ln (t^{11648})) + 531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.9

Multiply $t$ by $t^{7}$ by adding the exponents.

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Step 16.1.17.15.9.1

Move $t^{7}$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (1594323 (t^{7} t) e^{8 t} \cdot 224 + 1594323 t e^{8 t} (- \ln (t^{11648})) + 531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.9.2

Multiply $t^{7}$ by $t$ .

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Step 16.1.17.15.9.2.1

Raise $t$ to the power of $1$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (1594323 (t^{7} t^{1}) e^{8 t} \cdot 224 + 1594323 t e^{8 t} (- \ln (t^{11648})) + 531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.9.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (1594323 t^{7 + 1} e^{8 t} \cdot 224 + 1594323 t e^{8 t} (- \ln (t^{11648})) + 531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.9.3

Add $7$ and $1$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (1594323 t^{8} e^{8 t} \cdot 224 + 1594323 t e^{8 t} (- \ln (t^{11648})) + 531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.10

Multiply $1594323 t e^{8 t} (- \ln (t^{11648}))$ .

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Step 16.1.17.15.10.1

Multiply $- 1$ by $1594323$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (1594323 t^{8} e^{8 t} \cdot 224 - 1594323 t e^{8 t} \ln (t^{11648}) + 531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.10.2

Simplify $- 1594323 \ln (t^{11648})$ by moving $1594323$ inside the logarithm.

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (1594323 t^{8} e^{8 t} \cdot 224 + t e^{8 t} (- \ln ({(t^{11648})}^{1594323})) + 531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.11

Simplify each term.

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Step 16.1.17.15.11.1

Multiply $224$ by $1594323$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} + t e^{8 t} (- \ln ({(t^{11648})}^{1594323})) + 531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.11.2

Rewrite using the commutative property of multiplication.

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln ({(t^{11648})}^{1594323}) + 531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.11.3

Multiply the exponents in ${(t^{11648})}^{1594323}$ .

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Step 16.1.17.15.11.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{11648 \cdot 1594323}) + 531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.11.3.2

Multiply $11648$ by $1594323$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) + 531441 (1 + 2 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.12

Apply the distributive property.

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) + (531441 \cdot 1 + 531441 (2 e^{8 t})) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.13

Multiply $531441$ by $1$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) + (531441 + 531441 (2 e^{8 t})) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.14

Multiply $2$ by $531441$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) + (531441 + 1062882 e^{8 t}) (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.15

Expand $(531441 + 1062882 e^{8 t}) (- 7 t^{7} + 52)$ using the FOIL Method.

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Step 16.1.17.15.15.1

Apply the distributive property.

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) + 531441 (- 7 t^{7} + 52) + 1062882 e^{8 t} (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.15.2

Apply the distributive property.

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) + 531441 (- 7 t^{7}) + 531441 \cdot 52 + 1062882 e^{8 t} (- 7 t^{7} + 52)))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.15.3

Apply the distributive property.

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) + 531441 (- 7 t^{7}) + 531441 \cdot 52 + 1062882 e^{8 t} (- 7 t^{7}) + 1062882 e^{8 t} \cdot 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.16

Simplify each term.

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Step 16.1.17.15.16.1

Multiply $- 7$ by $531441$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) - 3720087 t^{7} + 531441 \cdot 52 + 1062882 e^{8 t} (- 7 t^{7}) + 1062882 e^{8 t} \cdot 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.16.2

Multiply $531441$ by $52$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) - 3720087 t^{7} + 27634932 + 1062882 e^{8 t} (- 7 t^{7}) + 1062882 e^{8 t} \cdot 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.16.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) - 3720087 t^{7} + 27634932 + 1062882 \cdot - 7 e^{8 t} t^{7} + 1062882 e^{8 t} \cdot 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.16.4

Multiply $1062882$ by $- 7$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) - 3720087 t^{7} + 27634932 - 7440174 e^{8 t} t^{7} + 1062882 e^{8 t} \cdot 52))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.1.17.15.16.5

Multiply $52$ by $1062882$ .

$\frac{\frac{3 ({(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) - 3720087 t^{7} + 27634932 - 7440174 e^{8 t} t^{7} + 55269864 e^{8 t}))}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

$\frac{\frac{3 {(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) - 3720087 t^{7} + 27634932 - 7440174 e^{8 t} t^{7} + 55269864 e^{8 t})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.2

Combine terms.

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Step 16.2.1

Rewrite $\frac{\frac{3 {(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) - 3720087 t^{7} + 27634932 - 7440174 e^{8 t} t^{7} + 55269864 e^{8 t})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$ as a product.

$\frac{3 {(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) - 3720087 t^{7} + 27634932 - 7440174 e^{8 t} t^{7} + 55269864 e^{8 t})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}} \cdot \frac{1}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$

Step 16.2.2

Multiply $\frac{3 {(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) - 3720087 t^{7} + 27634932 - 7440174 e^{8 t} t^{7} + 55269864 e^{8 t})}{t {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$ by $\frac{1}{{(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}}$ .

Step 16.2.3

Multiply ${(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}$ by ${(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}$ by adding the exponents.

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Step 16.2.3.1

Move ${(t^{7} - \ln (t^{52}))}^{\frac{2}{3}}$ .

$\frac{3 {(1 + 2 e^{8 t})}^{13} (357128352 t^{8} e^{8 t} - t e^{8 t} \ln (t^{18570674304}) - 3720087 t^{7} + 27634932 - 7440174 e^{8 t} t^{7} + 55269864 e^{8 t})}{t ({(t^{7} - \ln (t^{52}))}^{\frac{2}{3}} {(t^{7} - \ln (t^{52}))}^{\frac{2}{3}})}$

Step 16.2.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Step 16.2.3.3

Combine the numerators over the common denominator.

Step 16.2.3.4

Add $2$ and $2$ .