Calculus Examples

$y = 5 \sqrt[3]{- 9 x - 1}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{- 9 x - 1}$ as ${(- 9 x - 1)}^{\frac{1}{3}}$ .

$y = 5 {(- 9 x - 1)}^{\frac{1}{3}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (5 {(- 9 x - 1)}^{\frac{1}{3}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Since $5$ is constant with respect to $x$ , the derivative of $5 {(- 9 x - 1)}^{\frac{1}{3}}$ with respect to $x$ is $5 \frac{d}{d x} [{(- 9 x - 1)}^{\frac{1}{3}}]$ .

$5 \frac{d}{d x} [{(- 9 x - 1)}^{\frac{1}{3}}]$

Step 4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = - 9 x - 1$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $- 9 x - 1$ .

$5 (\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [- 9 x - 1])$

Step 4.2.2

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

$5 (\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [- 9 x - 1])$

Step 4.2.3

Replace all occurrences of $u$ with $- 9 x - 1$ .

$5 (\frac{1}{3} {(- 9 x - 1)}^{\frac{1}{3} - 1} \frac{d}{d x} [- 9 x - 1])$

Step 4.3

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

$5 (\frac{1}{3} {(- 9 x - 1)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [- 9 x - 1])$

Step 4.4

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

$5 (\frac{1}{3} {(- 9 x - 1)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [- 9 x - 1])$

Step 4.5

Combine the numerators over the common denominator.

$5 (\frac{1}{3} {(- 9 x - 1)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [- 9 x - 1])$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $3$ .

$5 (\frac{1}{3} {(- 9 x - 1)}^{\frac{1 - 3}{3}} \frac{d}{d x} [- 9 x - 1])$

Step 4.6.2

Subtract $3$ from $1$ .

$5 (\frac{1}{3} {(- 9 x - 1)}^{\frac{- 2}{3}} \frac{d}{d x} [- 9 x - 1])$

Step 4.7

Combine fractions.

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Step 4.7.1

Move the negative in front of the fraction.

$5 (\frac{1}{3} {(- 9 x - 1)}^{- \frac{2}{3}} \frac{d}{d x} [- 9 x - 1])$

Step 4.7.2

Combine $\frac{1}{3}$ and ${(- 9 x - 1)}^{- \frac{2}{3}}$ .

$5 (\frac{{(- 9 x - 1)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [- 9 x - 1])$

Step 4.7.3

Move ${(- 9 x - 1)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$5 (\frac{1}{3 {(- 9 x - 1)}^{\frac{2}{3}}} \frac{d}{d x} [- 9 x - 1])$

Step 4.7.4

Combine $\frac{1}{3 {(- 9 x - 1)}^{\frac{2}{3}}}$ and $5$ .

$\frac{5}{3 {(- 9 x - 1)}^{\frac{2}{3}}} \frac{d}{d x} [- 9 x - 1]$

Step 4.8

By the Sum Rule, the derivative of $- 9 x - 1$ with respect to $x$ is $\frac{d}{d x} [- 9 x] + \frac{d}{d x} [- 1]$ .

$\frac{5}{3 {(- 9 x - 1)}^{\frac{2}{3}}} (\frac{d}{d x} [- 9 x] + \frac{d}{d x} [- 1])$

Step 4.9

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$\frac{5}{3 {(- 9 x - 1)}^{\frac{2}{3}}} (- 9 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 4.10

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

$\frac{5}{3 {(- 9 x - 1)}^{\frac{2}{3}}} (- 9 \cdot 1 + \frac{d}{d x} [- 1])$

Step 4.11

Multiply $- 9$ by $1$ .

$\frac{5}{3 {(- 9 x - 1)}^{\frac{2}{3}}} (- 9 + \frac{d}{d x} [- 1])$

Step 4.12

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{5}{3 {(- 9 x - 1)}^{\frac{2}{3}}} (- 9 + 0)$

Step 4.13

Simplify terms.

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Step 4.13.1

Add $- 9$ and $0$ .

$\frac{5}{3 {(- 9 x - 1)}^{\frac{2}{3}}} \cdot - 9$

Step 4.13.2

Combine $\frac{5}{3 {(- 9 x - 1)}^{\frac{2}{3}}}$ and $- 9$ .

$\frac{5 \cdot - 9}{3 {(- 9 x - 1)}^{\frac{2}{3}}}$

Step 4.13.3

Multiply $5$ by $- 9$ .

$\frac{- 45}{3 {(- 9 x - 1)}^{\frac{2}{3}}}$

Step 4.13.4

Factor $3$ out of $- 45$ .

$\frac{3 \cdot - 15}{3 {(- 9 x - 1)}^{\frac{2}{3}}}$

Step 4.14

Cancel the common factors.

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Step 4.14.1

Factor $3$ out of $3 {(- 9 x - 1)}^{\frac{2}{3}}$ .

$\frac{3 \cdot - 15}{3 ({(- 9 x - 1)}^{\frac{2}{3}})}$

Step 4.14.2

Cancel the common factor.

$\frac{3 \cdot - 15}{3 {(- 9 x - 1)}^{\frac{2}{3}}}$

Step 4.14.3

Rewrite the expression.

$\frac{- 15}{{(- 9 x - 1)}^{\frac{2}{3}}}$

Step 4.15

Move the negative in front of the fraction.

$- \frac{15}{{(- 9 x - 1)}^{\frac{2}{3}}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{15}{{(- 9 x - 1)}^{\frac{2}{3}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{15}{{(- 9 x - 1)}^{\frac{2}{3}}}$