Calculus Examples

$h (t) = \sqrt[7]{t} - 7 e^{t}$

Step 1

Find the first derivative.

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Step 1.1

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

$\frac{d}{d t} [\sqrt[7]{t}] + \frac{d}{d t} [- 7 e^{t}]$

Step 1.2

Evaluate $\frac{d}{d t} [\sqrt[7]{t}]$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{t}$ as $t^{\frac{1}{7}}$ .

$\frac{d}{d t} [t^{\frac{1}{7}}] + \frac{d}{d t} [- 7 e^{t}]$

Step 1.2.2

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

$\frac{1}{7} t^{\frac{1}{7} - 1} + \frac{d}{d t} [- 7 e^{t}]$

Step 1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{1}{7} t^{\frac{1}{7} - 1 \cdot \frac{7}{7}} + \frac{d}{d t} [- 7 e^{t}]$

Step 1.2.4

Combine $- 1$ and $\frac{7}{7}$ .

$\frac{1}{7} t^{\frac{1}{7} + \frac{- 1 \cdot 7}{7}} + \frac{d}{d t} [- 7 e^{t}]$

Step 1.2.5

Combine the numerators over the common denominator.

$\frac{1}{7} t^{\frac{1 - 1 \cdot 7}{7}} + \frac{d}{d t} [- 7 e^{t}]$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $7$ .

$\frac{1}{7} t^{\frac{1 - 7}{7}} + \frac{d}{d t} [- 7 e^{t}]$

Step 1.2.6.2

Subtract $7$ from $1$ .

$\frac{1}{7} t^{\frac{- 6}{7}} + \frac{d}{d t} [- 7 e^{t}]$

Step 1.2.7

Move the negative in front of the fraction.

$\frac{1}{7} t^{- \frac{6}{7}} + \frac{d}{d t} [- 7 e^{t}]$

Step 1.3

Evaluate $\frac{d}{d t} [- 7 e^{t}]$ .

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Step 1.3.1

Since $- 7$ is constant with respect to $t$ , the derivative of $- 7 e^{t}$ with respect to $t$ is $- 7 \frac{d}{d t} [e^{t}]$ .

$\frac{1}{7} t^{- \frac{6}{7}} - 7 \frac{d}{d t} [e^{t}]$

Step 1.3.2

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

$\frac{1}{7} t^{- \frac{6}{7}} - 7 e^{t}$

Step 1.4

Simplify.

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Step 1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{7} \cdot \frac{1}{t^{\frac{6}{7}}} - 7 e^{t}$

Step 1.4.2

Multiply $\frac{1}{7}$ by $\frac{1}{t^{\frac{6}{7}}}$ .

$f' (t) = \frac{1}{7 t^{\frac{6}{7}}} - 7 e^{t}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $\frac{1}{7 t^{\frac{6}{7}}} - 7 e^{t}$ with respect to $t$ is $\frac{d}{d t} [\frac{1}{7 t^{\frac{6}{7}}}] + \frac{d}{d t} [- 7 e^{t}]$ .

$\frac{d}{d t} [\frac{1}{7 t^{\frac{6}{7}}}] + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2

Evaluate $\frac{d}{d t} [\frac{1}{7 t^{\frac{6}{7}}}]$ .

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Step 2.2.1

Since $\frac{1}{7}$ is constant with respect to $t$ , the derivative of $\frac{1}{7 t^{\frac{6}{7}}}$ with respect to $t$ is $\frac{1}{7} \frac{d}{d t} [\frac{1}{t^{\frac{6}{7}}}]$ .

$\frac{1}{7} \frac{d}{d t} [\frac{1}{t^{\frac{6}{7}}}] + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.2

Rewrite $\frac{1}{t^{\frac{6}{7}}}$ as ${(t^{\frac{6}{7}})}^{- 1}$ .

$\frac{1}{7} \frac{d}{d t} [{(t^{\frac{6}{7}})}^{- 1}] + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 1}$ and $g (t) = t^{\frac{6}{7}}$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u$ as $t^{\frac{6}{7}}$ .

$\frac{1}{7} (\frac{d}{d u} [u^{- 1}] \frac{d}{d t} [t^{\frac{6}{7}}]) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.3.2

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

$\frac{1}{7} (- u^{- 2} \frac{d}{d t} [t^{\frac{6}{7}}]) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.3.3

Replace all occurrences of $u$ with $t^{\frac{6}{7}}$ .

$\frac{1}{7} (- {(t^{\frac{6}{7}})}^{- 2} \frac{d}{d t} [t^{\frac{6}{7}}]) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{6}{7}$ .

$\frac{1}{7} (- {(t^{\frac{6}{7}})}^{- 2} (\frac{6}{7} t^{\frac{6}{7} - 1})) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.5

Multiply the exponents in ${(t^{\frac{6}{7}})}^{- 2}$ .

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Step 2.2.5.1

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

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

Step 2.2.5.2

Multiply $\frac{6}{7} \cdot - 2$ .

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Step 2.2.5.2.1

Combine $\frac{6}{7}$ and $- 2$ .

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

Step 2.2.5.2.2

Multiply $6$ by $- 2$ .

$\frac{1}{7} (- t^{\frac{- 12}{7}} (\frac{6}{7} t^{\frac{6}{7} - 1})) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.5.3

Move the negative in front of the fraction.

$\frac{1}{7} (- t^{- \frac{12}{7}} (\frac{6}{7} t^{\frac{6}{7} - 1})) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{1}{7} (- t^{- \frac{12}{7}} (\frac{6}{7} t^{\frac{6}{7} - 1 \cdot \frac{7}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.7

Combine $- 1$ and $\frac{7}{7}$ .

$\frac{1}{7} (- t^{- \frac{12}{7}} (\frac{6}{7} t^{\frac{6}{7} + \frac{- 1 \cdot 7}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.8

Combine the numerators over the common denominator.

$\frac{1}{7} (- t^{- \frac{12}{7}} (\frac{6}{7} t^{\frac{6 - 1 \cdot 7}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.9

Simplify the numerator.

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Step 2.2.9.1

Multiply $- 1$ by $7$ .

$\frac{1}{7} (- t^{- \frac{12}{7}} (\frac{6}{7} t^{\frac{6 - 7}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.9.2

Subtract $7$ from $6$ .

$\frac{1}{7} (- t^{- \frac{12}{7}} (\frac{6}{7} t^{\frac{- 1}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.10

Move the negative in front of the fraction.

$\frac{1}{7} (- t^{- \frac{12}{7}} (\frac{6}{7} t^{- \frac{1}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.11

Combine $\frac{6}{7}$ and $t^{- \frac{1}{7}}$ .

$\frac{1}{7} (- t^{- \frac{12}{7}} \frac{6 t^{- \frac{1}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.12

Combine $\frac{6 t^{- \frac{1}{7}}}{7}$ and $t^{- \frac{12}{7}}$ .

$\frac{1}{7} (- \frac{6 t^{- \frac{1}{7}} t^{- \frac{12}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.13

Multiply $t^{- \frac{1}{7}}$ by $t^{- \frac{12}{7}}$ by adding the exponents.

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Step 2.2.13.1

Move $t^{- \frac{12}{7}}$ .

$\frac{1}{7} (- \frac{6 (t^{- \frac{12}{7}} t^{- \frac{1}{7}})}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.13.2

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

$\frac{1}{7} (- \frac{6 t^{- \frac{12}{7} - \frac{1}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.13.3

Combine the numerators over the common denominator.

$\frac{1}{7} (- \frac{6 t^{\frac{- 12 - 1}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.13.4

Subtract $1$ from $- 12$ .

$\frac{1}{7} (- \frac{6 t^{\frac{- 13}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.13.5

Move the negative in front of the fraction.

$\frac{1}{7} (- \frac{6 t^{- \frac{13}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.14

Move $t^{- \frac{13}{7}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{7} (- \frac{6}{7 t^{\frac{13}{7}}}) + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.15

Multiply $\frac{1}{7}$ by $\frac{6}{7 t^{\frac{13}{7}}}$ .

$- \frac{6}{7 (7 t^{\frac{13}{7}})} + \frac{d}{d t} [- 7 e^{t}]$

Step 2.2.16

Multiply $7$ by $7$ .

$- \frac{6}{49 t^{\frac{13}{7}}} + \frac{d}{d t} [- 7 e^{t}]$

Step 2.3

Evaluate $\frac{d}{d t} [- 7 e^{t}]$ .

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Step 2.3.1

Since $- 7$ is constant with respect to $t$ , the derivative of $- 7 e^{t}$ with respect to $t$ is $- 7 \frac{d}{d t} [e^{t}]$ .

$- \frac{6}{49 t^{\frac{13}{7}}} - 7 \frac{d}{d t} [e^{t}]$

Step 2.3.2

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

$f'' (t) = - \frac{6}{49 t^{\frac{13}{7}}} - 7 e^{t}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $- \frac{6}{49 t^{\frac{13}{7}}} - 7 e^{t}$ with respect to $t$ is $\frac{d}{d t} [- \frac{6}{49 t^{\frac{13}{7}}}] + \frac{d}{d t} [- 7 e^{t}]$ .

$\frac{d}{d t} [- \frac{6}{49 t^{\frac{13}{7}}}] + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2

Evaluate $\frac{d}{d t} [- \frac{6}{49 t^{\frac{13}{7}}}]$ .

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Step 3.2.1

Since $- \frac{6}{49}$ is constant with respect to $t$ , the derivative of $- \frac{6}{49 t^{\frac{13}{7}}}$ with respect to $t$ is $- \frac{6}{49} \frac{d}{d t} [\frac{1}{t^{\frac{13}{7}}}]$ .

$- \frac{6}{49} \frac{d}{d t} [\frac{1}{t^{\frac{13}{7}}}] + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.2

Rewrite $\frac{1}{t^{\frac{13}{7}}}$ as ${(t^{\frac{13}{7}})}^{- 1}$ .

$- \frac{6}{49} \frac{d}{d t} [{(t^{\frac{13}{7}})}^{- 1}] + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 1}$ and $g (t) = t^{\frac{13}{7}}$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u$ as $t^{\frac{13}{7}}$ .

$- \frac{6}{49} (\frac{d}{d u} [u^{- 1}] \frac{d}{d t} [t^{\frac{13}{7}}]) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.3.2

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

$- \frac{6}{49} (- u^{- 2} \frac{d}{d t} [t^{\frac{13}{7}}]) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.3.3

Replace all occurrences of $u$ with $t^{\frac{13}{7}}$ .

$- \frac{6}{49} (- {(t^{\frac{13}{7}})}^{- 2} \frac{d}{d t} [t^{\frac{13}{7}}]) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{13}{7}$ .

$- \frac{6}{49} (- {(t^{\frac{13}{7}})}^{- 2} (\frac{13}{7} t^{\frac{13}{7} - 1})) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.5

Multiply the exponents in ${(t^{\frac{13}{7}})}^{- 2}$ .

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Step 3.2.5.1

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

$- \frac{6}{49} (- t^{\frac{13}{7} \cdot - 2} (\frac{13}{7} t^{\frac{13}{7} - 1})) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.5.2

Multiply $\frac{13}{7} \cdot - 2$ .

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Step 3.2.5.2.1

Combine $\frac{13}{7}$ and $- 2$ .

$- \frac{6}{49} (- t^{\frac{13 \cdot - 2}{7}} (\frac{13}{7} t^{\frac{13}{7} - 1})) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.5.2.2

Multiply $13$ by $- 2$ .

$- \frac{6}{49} (- t^{\frac{- 26}{7}} (\frac{13}{7} t^{\frac{13}{7} - 1})) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.5.3

Move the negative in front of the fraction.

$- \frac{6}{49} (- t^{- \frac{26}{7}} (\frac{13}{7} t^{\frac{13}{7} - 1})) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- \frac{6}{49} (- t^{- \frac{26}{7}} (\frac{13}{7} t^{\frac{13}{7} - 1 \cdot \frac{7}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.7

Combine $- 1$ and $\frac{7}{7}$ .

$- \frac{6}{49} (- t^{- \frac{26}{7}} (\frac{13}{7} t^{\frac{13}{7} + \frac{- 1 \cdot 7}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.8

Combine the numerators over the common denominator.

$- \frac{6}{49} (- t^{- \frac{26}{7}} (\frac{13}{7} t^{\frac{13 - 1 \cdot 7}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.9

Simplify the numerator.

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Step 3.2.9.1

Multiply $- 1$ by $7$ .

$- \frac{6}{49} (- t^{- \frac{26}{7}} (\frac{13}{7} t^{\frac{13 - 7}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.9.2

Subtract $7$ from $13$ .

$- \frac{6}{49} (- t^{- \frac{26}{7}} (\frac{13}{7} t^{\frac{6}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.10

Combine $\frac{13}{7}$ and $t^{\frac{6}{7}}$ .

$- \frac{6}{49} (- t^{- \frac{26}{7}} \frac{13 t^{\frac{6}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.11

Combine $\frac{13 t^{\frac{6}{7}}}{7}$ and $t^{- \frac{26}{7}}$ .

$- \frac{6}{49} (- \frac{13 t^{\frac{6}{7}} t^{- \frac{26}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.12

Multiply $t^{\frac{6}{7}}$ by $t^{- \frac{26}{7}}$ by adding the exponents.

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Step 3.2.12.1

Move $t^{- \frac{26}{7}}$ .

$- \frac{6}{49} (- \frac{13 (t^{- \frac{26}{7}} t^{\frac{6}{7}})}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.12.2

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

$- \frac{6}{49} (- \frac{13 t^{- \frac{26}{7} + \frac{6}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.12.3

Combine the numerators over the common denominator.

$- \frac{6}{49} (- \frac{13 t^{\frac{- 26 + 6}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.12.4

Add $- 26$ and $6$ .

$- \frac{6}{49} (- \frac{13 t^{\frac{- 20}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.12.5

Move the negative in front of the fraction.

$- \frac{6}{49} (- \frac{13 t^{- \frac{20}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.13

Move $t^{- \frac{20}{7}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{6}{49} (- \frac{13}{7 t^{\frac{20}{7}}}) + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.14

Multiply $- 1$ by $- 1$ .

$1 (\frac{6}{49}) \frac{13}{7 t^{\frac{20}{7}}} + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.15

Multiply $\frac{6}{49}$ by $1$ .

$\frac{6}{49} \cdot \frac{13}{7 t^{\frac{20}{7}}} + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.16

Multiply $\frac{6}{49}$ by $\frac{13}{7 t^{\frac{20}{7}}}$ .

$\frac{6 \cdot 13}{49 (7 t^{\frac{20}{7}})} + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.17

Multiply $6$ by $13$ .

$\frac{78}{49 (7 t^{\frac{20}{7}})} + \frac{d}{d t} [- 7 e^{t}]$

Step 3.2.18

Multiply $7$ by $49$ .

$\frac{78}{343 t^{\frac{20}{7}}} + \frac{d}{d t} [- 7 e^{t}]$

Step 3.3

Evaluate $\frac{d}{d t} [- 7 e^{t}]$ .

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Step 3.3.1

Since $- 7$ is constant with respect to $t$ , the derivative of $- 7 e^{t}$ with respect to $t$ is $- 7 \frac{d}{d t} [e^{t}]$ .

$\frac{78}{343 t^{\frac{20}{7}}} - 7 \frac{d}{d t} [e^{t}]$

Step 3.3.2

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

$f''' (t) = \frac{78}{343 t^{\frac{20}{7}}} - 7 e^{t}$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $\frac{78}{343 t^{\frac{20}{7}}} - 7 e^{t}$ with respect to $t$ is $\frac{d}{d t} [\frac{78}{343 t^{\frac{20}{7}}}] + \frac{d}{d t} [- 7 e^{t}]$ .

$\frac{d}{d t} [\frac{78}{343 t^{\frac{20}{7}}}] + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2

Evaluate $\frac{d}{d t} [\frac{78}{343 t^{\frac{20}{7}}}]$ .

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Step 4.2.1

Since $\frac{78}{343}$ is constant with respect to $t$ , the derivative of $\frac{78}{343 t^{\frac{20}{7}}}$ with respect to $t$ is $\frac{78}{343} \frac{d}{d t} [\frac{1}{t^{\frac{20}{7}}}]$ .

$\frac{78}{343} \frac{d}{d t} [\frac{1}{t^{\frac{20}{7}}}] + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.2

Rewrite $\frac{1}{t^{\frac{20}{7}}}$ as ${(t^{\frac{20}{7}})}^{- 1}$ .

$\frac{78}{343} \frac{d}{d t} [{(t^{\frac{20}{7}})}^{- 1}] + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 1}$ and $g (t) = t^{\frac{20}{7}}$ .

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Step 4.2.3.1

To apply the Chain Rule, set $u$ as $t^{\frac{20}{7}}$ .

$\frac{78}{343} (\frac{d}{d u} [u^{- 1}] \frac{d}{d t} [t^{\frac{20}{7}}]) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.3.2

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

$\frac{78}{343} (- u^{- 2} \frac{d}{d t} [t^{\frac{20}{7}}]) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.3.3

Replace all occurrences of $u$ with $t^{\frac{20}{7}}$ .

$\frac{78}{343} (- {(t^{\frac{20}{7}})}^{- 2} \frac{d}{d t} [t^{\frac{20}{7}}]) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{20}{7}$ .

$\frac{78}{343} (- {(t^{\frac{20}{7}})}^{- 2} (\frac{20}{7} t^{\frac{20}{7} - 1})) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.5

Multiply the exponents in ${(t^{\frac{20}{7}})}^{- 2}$ .

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Step 4.2.5.1

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

$\frac{78}{343} (- t^{\frac{20}{7} \cdot - 2} (\frac{20}{7} t^{\frac{20}{7} - 1})) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.5.2

Multiply $\frac{20}{7} \cdot - 2$ .

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Step 4.2.5.2.1

Combine $\frac{20}{7}$ and $- 2$ .

$\frac{78}{343} (- t^{\frac{20 \cdot - 2}{7}} (\frac{20}{7} t^{\frac{20}{7} - 1})) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.5.2.2

Multiply $20$ by $- 2$ .

$\frac{78}{343} (- t^{\frac{- 40}{7}} (\frac{20}{7} t^{\frac{20}{7} - 1})) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.5.3

Move the negative in front of the fraction.

$\frac{78}{343} (- t^{- \frac{40}{7}} (\frac{20}{7} t^{\frac{20}{7} - 1})) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{78}{343} (- t^{- \frac{40}{7}} (\frac{20}{7} t^{\frac{20}{7} - 1 \cdot \frac{7}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.7

Combine $- 1$ and $\frac{7}{7}$ .

$\frac{78}{343} (- t^{- \frac{40}{7}} (\frac{20}{7} t^{\frac{20}{7} + \frac{- 1 \cdot 7}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.8

Combine the numerators over the common denominator.

$\frac{78}{343} (- t^{- \frac{40}{7}} (\frac{20}{7} t^{\frac{20 - 1 \cdot 7}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.9

Simplify the numerator.

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Step 4.2.9.1

Multiply $- 1$ by $7$ .

$\frac{78}{343} (- t^{- \frac{40}{7}} (\frac{20}{7} t^{\frac{20 - 7}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.9.2

Subtract $7$ from $20$ .

$\frac{78}{343} (- t^{- \frac{40}{7}} (\frac{20}{7} t^{\frac{13}{7}})) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.10

Combine $\frac{20}{7}$ and $t^{\frac{13}{7}}$ .

$\frac{78}{343} (- t^{- \frac{40}{7}} \frac{20 t^{\frac{13}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.11

Combine $\frac{20 t^{\frac{13}{7}}}{7}$ and $t^{- \frac{40}{7}}$ .

$\frac{78}{343} (- \frac{20 t^{\frac{13}{7}} t^{- \frac{40}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.12

Multiply $t^{\frac{13}{7}}$ by $t^{- \frac{40}{7}}$ by adding the exponents.

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Step 4.2.12.1

Move $t^{- \frac{40}{7}}$ .

$\frac{78}{343} (- \frac{20 (t^{- \frac{40}{7}} t^{\frac{13}{7}})}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.12.2

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

$\frac{78}{343} (- \frac{20 t^{- \frac{40}{7} + \frac{13}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.12.3

Combine the numerators over the common denominator.

$\frac{78}{343} (- \frac{20 t^{\frac{- 40 + 13}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.12.4

Add $- 40$ and $13$ .

$\frac{78}{343} (- \frac{20 t^{\frac{- 27}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.12.5

Move the negative in front of the fraction.

$\frac{78}{343} (- \frac{20 t^{- \frac{27}{7}}}{7}) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.13

Move $t^{- \frac{27}{7}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{78}{343} (- \frac{20}{7 t^{\frac{27}{7}}}) + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.14

Multiply $\frac{78}{343}$ by $\frac{20}{7 t^{\frac{27}{7}}}$ .

$- \frac{78 \cdot 20}{343 (7 t^{\frac{27}{7}})} + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.15

Multiply $78$ by $20$ .

$- \frac{1560}{343 (7 t^{\frac{27}{7}})} + \frac{d}{d t} [- 7 e^{t}]$

Step 4.2.16

Multiply $7$ by $343$ .

$- \frac{1560}{2401 t^{\frac{27}{7}}} + \frac{d}{d t} [- 7 e^{t}]$

Step 4.3

Evaluate $\frac{d}{d t} [- 7 e^{t}]$ .

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Step 4.3.1

Since $- 7$ is constant with respect to $t$ , the derivative of $- 7 e^{t}$ with respect to $t$ is $- 7 \frac{d}{d t} [e^{t}]$ .

$- \frac{1560}{2401 t^{\frac{27}{7}}} - 7 \frac{d}{d t} [e^{t}]$

Step 4.3.2

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

$f^{4} (t) = - \frac{1560}{2401 t^{\frac{27}{7}}} - 7 e^{t}$

Step 5

The fourth derivative of $h (t)$ with respect to $t$ is $- \frac{1560}{2401 t^{\frac{27}{7}}} - 7 e^{t}$ .

$- \frac{1560}{2401 t^{\frac{27}{7}}} - 7 e^{t}$