Calculus Examples

$24 {(x^{2} + x)}^{\frac{2}{3}}$

Step 1

Since $24$ is constant with respect to $x$ , the derivative of $24 {(x^{2} + x)}^{\frac{2}{3}}$ with respect to $x$ is $24 \frac{d}{d x} [{(x^{2} + x)}^{\frac{2}{3}}]$ .

$24 \frac{d}{d x} [{(x^{2} + x)}^{\frac{2}{3}}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = x^{2} + x$ .

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Step 2.1

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

$24 (\frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [x^{2} + x])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{2}{3}$ .

$24 (\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [x^{2} + x])$

Step 2.3

Replace all occurrences of $u$ with $x^{2} + x$ .

$24 (\frac{2}{3} {(x^{2} + x)}^{\frac{2}{3} - 1} \frac{d}{d x} [x^{2} + x])$

Step 3

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

$24 (\frac{2}{3} {(x^{2} + x)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x^{2} + x])$

Step 4

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

$24 (\frac{2}{3} {(x^{2} + x)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [x^{2} + x])$

Step 5

Combine the numerators over the common denominator.

$24 (\frac{2}{3} {(x^{2} + x)}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d x} [x^{2} + x])$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

$24 (\frac{2}{3} {(x^{2} + x)}^{\frac{2 - 3}{3}} \frac{d}{d x} [x^{2} + x])$

Step 6.2

Subtract $3$ from $2$ .

$24 (\frac{2}{3} {(x^{2} + x)}^{\frac{- 1}{3}} \frac{d}{d x} [x^{2} + x])$

Step 7

Move the negative in front of the fraction.

$24 (\frac{2}{3} {(x^{2} + x)}^{- \frac{1}{3}} \frac{d}{d x} [x^{2} + x])$

Step 8

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

$24 (\frac{2 {(x^{2} + x)}^{- \frac{1}{3}}}{3} \frac{d}{d x} [x^{2} + x])$

Step 9

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

$24 (\frac{2}{3 {(x^{2} + x)}^{\frac{1}{3}}} \frac{d}{d x} [x^{2} + x])$

Step 10

Combine $\frac{2}{3 {(x^{2} + x)}^{\frac{1}{3}}}$ and $24$ .

$\frac{2 \cdot 24}{3 {(x^{2} + x)}^{\frac{1}{3}}} \frac{d}{d x} [x^{2} + x]$

Step 11

Multiply $2$ by $24$ .

$\frac{48}{3 {(x^{2} + x)}^{\frac{1}{3}}} \frac{d}{d x} [x^{2} + x]$

Step 12

Factor $3$ out of $48$ .

$\frac{3 \cdot 16}{3 {(x^{2} + x)}^{\frac{1}{3}}} \frac{d}{d x} [x^{2} + x]$

Step 13

Cancel the common factors.

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Step 13.1

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

$\frac{3 \cdot 16}{3 ({(x^{2} + x)}^{\frac{1}{3}})} \frac{d}{d x} [x^{2} + x]$

Step 13.2

Cancel the common factor.

$\frac{3 \cdot 16}{3 {(x^{2} + x)}^{\frac{1}{3}}} \frac{d}{d x} [x^{2} + x]$

Step 13.3

Rewrite the expression.

$\frac{16}{{(x^{2} + x)}^{\frac{1}{3}}} \frac{d}{d x} [x^{2} + x]$

Step 14

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

$\frac{16}{{(x^{2} + x)}^{\frac{1}{3}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [x])$

Step 15

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

$\frac{16}{{(x^{2} + x)}^{\frac{1}{3}}} (2 x + \frac{d}{d x} [x])$

Step 16

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

$\frac{16}{{(x^{2} + x)}^{\frac{1}{3}}} (2 x + 1)$

Step 17

Simplify.

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Step 17.1

Reorder the factors of $\frac{16}{{(x^{2} + x)}^{\frac{1}{3}}} (2 x + 1)$ .

$(2 x + 1) \frac{16}{{(x^{2} + x)}^{\frac{1}{3}}}$

Step 17.2

Multiply $2 x + 1$ by $\frac{16}{{(x^{2} + x)}^{\frac{1}{3}}}$ .

$\frac{(2 x + 1) \cdot 16}{{(x^{2} + x)}^{\frac{1}{3}}}$

Step 17.3

Move $16$ to the left of $2 x + 1$ .

$\frac{16 (2 x + 1)}{{(x^{2} + x)}^{\frac{1}{3}}}$