Calculus Examples

$g (v) = - 6 {(5 v - 3)}^{- 4} + 5 {(\ln (v^{3}))}^{\frac{6}{5}} - \sqrt[6]{1 + e^{2 x}}$

Step 1

By the Sum Rule, the derivative of $- 6 {(5 v - 3)}^{- 4} + 5 {(\ln (v^{3}))}^{\frac{6}{5}} - \sqrt[6]{1 + e^{2 x}}$ with respect to $v$ is $\frac{d}{d v} [- 6 {(5 v - 3)}^{- 4}] + \frac{d}{d v} [5 {(\ln (v^{3}))}^{\frac{6}{5}}] + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$ .

$\frac{d}{d v} [- 6 {(5 v - 3)}^{- 4}] + \frac{d}{d v} [5 {(\ln (v^{3}))}^{\frac{6}{5}}] + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 2

Evaluate $\frac{d}{d v} [- 6 {(5 v - 3)}^{- 4}]$ .

Tap for more steps...

Step 2.1

Since $- 6$ is constant with respect to $v$ , the derivative of $- 6 {(5 v - 3)}^{- 4}$ with respect to $v$ is $- 6 \frac{d}{d v} [{(5 v - 3)}^{- 4}]$ .

$- 6 \frac{d}{d v} [{(5 v - 3)}^{- 4}] + \frac{d}{d v} [5 {(\ln (v^{3}))}^{\frac{6}{5}}] + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d v} [f (g (v))]$ is $f' (g (v)) g' (v)$ where $f (v) = v^{- 4}$ and $g (v) = 5 v - 3$ .

Tap for more steps...

Step 2.2.1

To apply the Chain Rule, set $u_{1}$ as $5 v - 3$ .

$- 6 (\frac{d}{d u_{1}} [{u_{1}}^{- 4}] \frac{d}{d v} [5 v - 3]) + \frac{d}{d v} [5 {(\ln (v^{3}))}^{\frac{6}{5}}] + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = - 4$ .

$- 6 (- 4 {u_{1}}^{- 5} \frac{d}{d v} [5 v - 3]) + \frac{d}{d v} [5 {(\ln (v^{3}))}^{\frac{6}{5}}] + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 2.2.3

Replace all occurrences of $u_{1}$ with $5 v - 3$ .

$- 6 (- 4 {(5 v - 3)}^{- 5} \frac{d}{d v} [5 v - 3]) + \frac{d}{d v} [5 {(\ln (v^{3}))}^{\frac{6}{5}}] + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 2.3

By the Sum Rule, the derivative of $5 v - 3$ with respect to $v$ is $\frac{d}{d v} [5 v] + \frac{d}{d v} [- 3]$ .

$- 6 (- 4 {(5 v - 3)}^{- 5} (\frac{d}{d v} [5 v] + \frac{d}{d v} [- 3])) + \frac{d}{d v} [5 {(\ln (v^{3}))}^{\frac{6}{5}}] + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 2.4

Since $5$ is constant with respect to $v$ , the derivative of $5 v$ with respect to $v$ is $5 \frac{d}{d v} [v]$ .

$- 6 (- 4 {(5 v - 3)}^{- 5} (5 \frac{d}{d v} [v] + \frac{d}{d v} [- 3])) + \frac{d}{d v} [5 {(\ln (v^{3}))}^{\frac{6}{5}}] + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 2.5

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

$- 6 (- 4 {(5 v - 3)}^{- 5} (5 \cdot 1 + \frac{d}{d v} [- 3])) + \frac{d}{d v} [5 {(\ln (v^{3}))}^{\frac{6}{5}}] + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 2.6

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

$- 6 (- 4 {(5 v - 3)}^{- 5} (5 \cdot 1 + 0)) + \frac{d}{d v} [5 {(\ln (v^{3}))}^{\frac{6}{5}}] + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 2.7

Multiply $5$ by $1$ .

$- 6 (- 4 {(5 v - 3)}^{- 5} (5 + 0)) + \frac{d}{d v} [5 {(\ln (v^{3}))}^{\frac{6}{5}}] + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 2.8

Add $5$ and $0$ .

$- 6 (- 4 {(5 v - 3)}^{- 5} \cdot 5) + \frac{d}{d v} [5 {(\ln (v^{3}))}^{\frac{6}{5}}] + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 2.9

Multiply $5$ by $- 4$ .

$- 6 (- 20 {(5 v - 3)}^{- 5}) + \frac{d}{d v} [5 {(\ln (v^{3}))}^{\frac{6}{5}}] + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 2.10

Multiply $- 20$ by $- 6$ .

$120 {(5 v - 3)}^{- 5} + \frac{d}{d v} [5 {(\ln (v^{3}))}^{\frac{6}{5}}] + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3

Evaluate $\frac{d}{d v} [5 {(\ln (v^{3}))}^{\frac{6}{5}}]$ .

Tap for more steps...

Step 3.1

Since $5$ is constant with respect to $v$ , the derivative of $5 {\ln (v^{3})}^{\frac{6}{5}}$ with respect to $v$ is $5 \frac{d}{d v} [{\ln (v^{3})}^{\frac{6}{5}}]$ .

$120 {(5 v - 3)}^{- 5} + 5 \frac{d}{d v} [{\ln (v^{3})}^{\frac{6}{5}}] + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d v} [f (g (v))]$ is $f' (g (v)) g' (v)$ where $f (v) = v^{\frac{6}{5}}$ and $g (v) = \ln (v^{3})$ .

Tap for more steps...

Step 3.2.1

To apply the Chain Rule, set $u_{2}$ as $\ln (v^{3})$ .

$120 {(5 v - 3)}^{- 5} + 5 (\frac{d}{d u_{2}} [{u_{2}}^{\frac{6}{5}}] \frac{d}{d v} [\ln (v^{3})]) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{6}{5}$ .

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {u_{2}}^{\frac{6}{5} - 1} \frac{d}{d v} [\ln (v^{3})]) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.2.3

Replace all occurrences of $u_{2}$ with $\ln (v^{3})$ .

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{6}{5} - 1} \frac{d}{d v} [\ln (v^{3})]) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d v} [f (g (v))]$ is $f' (g (v)) g' (v)$ where $f (v) = \ln (v)$ and $g (v) = v^{3}$ .

Tap for more steps...

Step 3.3.1

To apply the Chain Rule, set $u_{3}$ as $v^{3}$ .

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{6}{5} - 1} (\frac{d}{d u_{3}} [\ln (u_{3})] \frac{d}{d v} [v^{3}])) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.3.2

The derivative of $\ln (u_{3})$ with respect to $u_{3}$ is $\frac{1}{u_{3}}$ .

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{6}{5} - 1} (\frac{1}{u_{3}} \frac{d}{d v} [v^{3}])) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.3.3

Replace all occurrences of $u_{3}$ with $v^{3}$ .

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{6}{5} - 1} (\frac{1}{v^{3}} \frac{d}{d v} [v^{3}])) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.4

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

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{6}{5} - 1} (\frac{1}{v^{3}} (3 v^{2}))) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{6}{5} - 1 \cdot \frac{5}{5}} (\frac{1}{v^{3}} (3 v^{2}))) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.6

Combine $- 1$ and $\frac{5}{5}$ .

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{6}{5} + \frac{- 1 \cdot 5}{5}} (\frac{1}{v^{3}} (3 v^{2}))) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.7

Combine the numerators over the common denominator.

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{6 - 1 \cdot 5}{5}} (\frac{1}{v^{3}} (3 v^{2}))) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.8

Simplify the numerator.

Tap for more steps...

Step 3.8.1

Multiply $- 1$ by $5$ .

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{6 - 5}{5}} (\frac{1}{v^{3}} (3 v^{2}))) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.8.2

Subtract $5$ from $6$ .

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{1}{5}} (\frac{1}{v^{3}} (3 v^{2}))) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.9

Combine $3$ and $\frac{1}{v^{3}}$ .

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{1}{5}} (\frac{3}{v^{3}} v^{2})) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.10

Combine $\frac{3}{v^{3}}$ and $v^{2}$ .

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{1}{5}} \frac{3 v^{2}}{v^{3}}) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.11

Cancel the common factor of $v^{2}$ and $v^{3}$ .

Tap for more steps...

Step 3.11.1

Factor $v^{2}$ out of $3 v^{2}$ .

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{1}{5}} \frac{v^{2} \cdot 3}{v^{3}}) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.11.2

Cancel the common factors.

Tap for more steps...

Step 3.11.2.1

Factor $v^{2}$ out of $v^{3}$ .

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{1}{5}} \frac{v^{2} \cdot 3}{v^{2} v}) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.11.2.2

Cancel the common factor.

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{1}{5}} \frac{v^{2} \cdot 3}{v^{2} v}) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.11.2.3

Rewrite the expression.

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6}{5} {\ln (v^{3})}^{\frac{1}{5}} \frac{3}{v}) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.12

Combine $\frac{6}{5}$ and ${\ln (v^{3})}^{\frac{1}{5}}$ .

$120 {(5 v - 3)}^{- 5} + 5 (\frac{6 {\ln (v^{3})}^{\frac{1}{5}}}{5} \cdot \frac{3}{v}) + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.13

Multiply $\frac{6 {\ln (v^{3})}^{\frac{1}{5}}}{5}$ by $\frac{3}{v}$ .

$120 {(5 v - 3)}^{- 5} + 5 \frac{6 {\ln (v^{3})}^{\frac{1}{5}} \cdot 3}{5 v} + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.14

Multiply $3$ by $6$ .

$120 {(5 v - 3)}^{- 5} + 5 \frac{18 {\ln (v^{3})}^{\frac{1}{5}}}{5 v} + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.15

Combine $5$ and $\frac{18 {\ln (v^{3})}^{\frac{1}{5}}}{5 v}$ .

$120 {(5 v - 3)}^{- 5} + \frac{5 (18 {\ln (v^{3})}^{\frac{1}{5}})}{5 v} + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.16

Multiply $18$ by $5$ .

$120 {(5 v - 3)}^{- 5} + \frac{90 {\ln (v^{3})}^{\frac{1}{5}}}{5 v} + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.17

Factor $5$ out of $90 {\ln (v^{3})}^{\frac{1}{5}}$ .

$120 {(5 v - 3)}^{- 5} + \frac{5 (18 {\ln (v^{3})}^{\frac{1}{5}})}{5 v} + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.18

Cancel the common factors.

Tap for more steps...

Step 3.18.1

Factor $5$ out of $5 v$ .

$120 {(5 v - 3)}^{- 5} + \frac{5 (18 {\ln (v^{3})}^{\frac{1}{5}})}{5 (v)} + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.18.2

Cancel the common factor.

$120 {(5 v - 3)}^{- 5} + \frac{5 (18 {\ln (v^{3})}^{\frac{1}{5}})}{5 v} + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 3.18.3

Rewrite the expression.

$120 {(5 v - 3)}^{- 5} + \frac{18 {\ln (v^{3})}^{\frac{1}{5}}}{v} + \frac{d}{d v} [- \sqrt[6]{1 + e^{2 x}}]$

Step 4

Since $- \sqrt[6]{1 + e^{2 x}}$ is constant with respect to $v$ , the derivative of $- \sqrt[6]{1 + e^{2 x}}$ with respect to $v$ is $0$ .

$120 {(5 v - 3)}^{- 5} + \frac{18 {\ln (v^{3})}^{\frac{1}{5}}}{v} + 0$

Step 5

Simplify.

Tap for more steps...

Step 5.1

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

$120 \frac{1}{{(5 v - 3)}^{5}} + \frac{18 {\ln (v^{3})}^{\frac{1}{5}}}{v} + 0$

Step 5.2

Combine terms.

Tap for more steps...

Step 5.2.1

Combine $120$ and $\frac{1}{{(5 v - 3)}^{5}}$ .

$\frac{120}{{(5 v - 3)}^{5}} + \frac{18 {\ln (v^{3})}^{\frac{1}{5}}}{v} + 0$

Step 5.2.2

Add $\frac{120}{{(5 v - 3)}^{5}} + \frac{18 {\ln (v^{3})}^{\frac{1}{5}}}{v}$ and $0$ .

$\frac{120}{{(5 v - 3)}^{5}} + \frac{18 {\ln (v^{3})}^{\frac{1}{5}}}{v}$

Step 5.3

Reorder terms.

$\frac{18 {\ln (v^{3})}^{\frac{1}{5}}}{v} + \frac{120}{{(5 v - 3)}^{5}}$