Calculus Examples

$y = \sqrt[3]{- 9 x^{2} - 4 x}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{- 9 x^{2} - 4 x}$ as ${(- 9 x^{2} - 4 x)}^{\frac{1}{3}}$ .

$y = {(- 9 x^{2} - 4 x)}^{\frac{1}{3}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(- 9 x^{2} - 4 x)}^{\frac{1}{3}})$

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

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 $g (x) = - 9 x^{2} - 4 x$ .

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Step 4.1.1

To apply the Chain Rule, set $u$ as $- 9 x^{2} - 4 x$ .

$\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [- 9 x^{2} - 4 x]$

Step 4.1.2

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

$\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [- 9 x^{2} - 4 x]$

Step 4.1.3

Replace all occurrences of $u$ with $- 9 x^{2} - 4 x$ .

$\frac{1}{3} {(- 9 x^{2} - 4 x)}^{\frac{1}{3} - 1} \frac{d}{d x} [- 9 x^{2} - 4 x]$

Step 4.2

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

$\frac{1}{3} {(- 9 x^{2} - 4 x)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [- 9 x^{2} - 4 x]$

Step 4.3

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

$\frac{1}{3} {(- 9 x^{2} - 4 x)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [- 9 x^{2} - 4 x]$

Step 4.4

Combine the numerators over the common denominator.

$\frac{1}{3} {(- 9 x^{2} - 4 x)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [- 9 x^{2} - 4 x]$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} {(- 9 x^{2} - 4 x)}^{\frac{1 - 3}{3}} \frac{d}{d x} [- 9 x^{2} - 4 x]$

Step 4.5.2

Subtract $3$ from $1$ .

$\frac{1}{3} {(- 9 x^{2} - 4 x)}^{\frac{- 2}{3}} \frac{d}{d x} [- 9 x^{2} - 4 x]$

Step 4.6

Combine fractions.

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Step 4.6.1

Move the negative in front of the fraction.

$\frac{1}{3} {(- 9 x^{2} - 4 x)}^{- \frac{2}{3}} \frac{d}{d x} [- 9 x^{2} - 4 x]$

Step 4.6.2

Combine $\frac{1}{3}$ and ${(- 9 x^{2} - 4 x)}^{- \frac{2}{3}}$ .

$\frac{{(- 9 x^{2} - 4 x)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [- 9 x^{2} - 4 x]$

Step 4.6.3

Move ${(- 9 x^{2} - 4 x)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}} \frac{d}{d x} [- 9 x^{2} - 4 x]$

Step 4.7

By the Sum Rule, the derivative of $- 9 x^{2} - 4 x$ with respect to $x$ is $\frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 x]$ .

$\frac{1}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}} (\frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 4 x])$

Step 4.8

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x^{2}$ with respect to $x$ is $- 9 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}} (- 9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x])$

Step 4.9

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

$\frac{1}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}} (- 9 (2 x) + \frac{d}{d x} [- 4 x])$

Step 4.10

Multiply $2$ by $- 9$ .

$\frac{1}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}} (- 18 x + \frac{d}{d x} [- 4 x])$

Step 4.11

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$\frac{1}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}} (- 18 x - 4 \frac{d}{d x} [x])$

Step 4.12

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

$\frac{1}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}} (- 18 x - 4 \cdot 1)$

Step 4.13

Multiply $- 4$ by $1$ .

$\frac{1}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}} (- 18 x - 4)$

Step 4.14

Simplify.

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Step 4.14.1

Reorder the factors of $\frac{1}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}} (- 18 x - 4)$ .

$(- 18 x - 4) \frac{1}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}}$

Step 4.14.2

Multiply $- 18 x - 4$ by $\frac{1}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}}$ .

$\frac{- 18 x - 4}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}}$

Step 4.14.3

Factor $2$ out of $- 18 x - 4$ .

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Step 4.14.3.1

Factor $2$ out of $- 18 x$ .

$\frac{2 (- 9 x) - 4}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}}$

Step 4.14.3.2

Factor $2$ out of $- 4$ .

$\frac{2 (- 9 x) + 2 (- 2)}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}}$

Step 4.14.3.3

Factor $2$ out of $2 (- 9 x) + 2 (- 2)$ .

$\frac{2 (- 9 x - 2)}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}}$

Step 4.14.4

Factor $- 1$ out of $- 9 x$ .

$\frac{2 (- (9 x) - 2)}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}}$

Step 4.14.5

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{2 (- (9 x) - 1 (2))}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}}$

Step 4.14.6

Factor $- 1$ out of $- (9 x) - 1 (2)$ .

$\frac{2 (- (9 x + 2))}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}}$

Step 4.14.7

Rewrite $- (9 x + 2)$ as $- 1 (9 x + 2)$ .

$\frac{2 (- 1 (9 x + 2))}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}}$

Step 4.14.8

Move the negative in front of the fraction.

$- \frac{2 (9 x + 2)}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}}$

Step 5

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

$y' = - \frac{2 (9 x + 2)}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}}$

Step 6

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

$\frac{d y}{d x} = - \frac{2 (9 x + 2)}{3 {(- 9 x^{2} - 4 x)}^{\frac{2}{3}}}$