Calculus Examples

$\sqrt[15]{7 x + 5}$

Step 1

This derivative could not be completed using the product rule. Mathway will use another method.

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[15]{7 x + 5}$ as ${(7 x + 5)}^{\frac{1}{15}}$ .

$\frac{d}{d x} [{(7 x + 5)}^{\frac{1}{15}}]$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{15}}$ and $g (x) = 7 x + 5$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $7 x + 5$ .

$\frac{d}{d u} [u^{\frac{1}{15}}] \frac{d}{d x} [7 x + 5]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{15}$ .

$\frac{1}{15} u^{\frac{1}{15} - 1} \frac{d}{d x} [7 x + 5]$

Step 3.3

Replace all occurrences of $u$ with $7 x + 5$ .

$\frac{1}{15} {(7 x + 5)}^{\frac{1}{15} - 1} \frac{d}{d x} [7 x + 5]$

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{1}{15} {(7 x + 5)}^{\frac{1}{15} - 1 \cdot \frac{15}{15}} \frac{d}{d x} [7 x + 5]$

Step 5

Combine $- 1$ and $\frac{15}{15}$ .

$\frac{1}{15} {(7 x + 5)}^{\frac{1}{15} + \frac{- 1 \cdot 15}{15}} \frac{d}{d x} [7 x + 5]$

Step 6

Combine the numerators over the common denominator.

$\frac{1}{15} {(7 x + 5)}^{\frac{1 - 1 \cdot 15}{15}} \frac{d}{d x} [7 x + 5]$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $15$ .

$\frac{1}{15} {(7 x + 5)}^{\frac{1 - 15}{15}} \frac{d}{d x} [7 x + 5]$

Step 7.2

Subtract $15$ from $1$ .

$\frac{1}{15} {(7 x + 5)}^{\frac{- 14}{15}} \frac{d}{d x} [7 x + 5]$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$\frac{1}{15} {(7 x + 5)}^{- \frac{14}{15}} \frac{d}{d x} [7 x + 5]$

Step 8.2

Combine $\frac{1}{15}$ and ${(7 x + 5)}^{- \frac{14}{15}}$ .

$\frac{{(7 x + 5)}^{- \frac{14}{15}}}{15} \frac{d}{d x} [7 x + 5]$

Step 8.3

Move ${(7 x + 5)}^{- \frac{14}{15}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{15 {(7 x + 5)}^{\frac{14}{15}}} \frac{d}{d x} [7 x + 5]$

Step 9

By the Sum Rule, the derivative of $7 x + 5$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [5]$ .

$\frac{1}{15 {(7 x + 5)}^{\frac{14}{15}}} (\frac{d}{d x} [7 x] + \frac{d}{d x} [5])$

Step 10

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

$\frac{1}{15 {(7 x + 5)}^{\frac{14}{15}}} (7 \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 11

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

$\frac{1}{15 {(7 x + 5)}^{\frac{14}{15}}} (7 \cdot 1 + \frac{d}{d x} [5])$

Step 12

Multiply $7$ by $1$ .

$\frac{1}{15 {(7 x + 5)}^{\frac{14}{15}}} (7 + \frac{d}{d x} [5])$

Step 13

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{1}{15 {(7 x + 5)}^{\frac{14}{15}}} (7 + 0)$

Step 14

Combine fractions.

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Step 14.1

Add $7$ and $0$ .

$\frac{1}{15 {(7 x + 5)}^{\frac{14}{15}}} \cdot 7$

Step 14.2

Combine $\frac{1}{15 {(7 x + 5)}^{\frac{14}{15}}}$ and $7$ .

$\frac{7}{15 {(7 x + 5)}^{\frac{14}{15}}}$