Calculus Examples

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

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

Step 1

Since

\frac{2}{3}

$\frac{2}{3}$ is constant with respect to

x

$x$ , the derivative of

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

$\frac{2}{3} {(x^{2} + 1)}^{\frac{3}{2}}$ with respect to

x

$x$ is

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

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

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

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

Step 2

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = x^{\frac{3}{2}}$ and

$g (x) = x^{2} + 1$ .

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Step 2.1

To apply the Chain Rule, set

$u$ as

$x^{2} + 1$ .

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

Step 2.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

$n = \frac{3}{2}$ .

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

Step 2.3

Replace all occurrences of

$u$ with

$x^{2} + 1$ .

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

Step 3

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 4

Combine

$- 1$ and

$\frac{2}{2}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply

$- 1$ by

$2$ .

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

Step 6.2

Subtract

$2$ from

$3$ .

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

Step 7

Simplify terms.

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Step 7.1

Combine

$\frac{3}{2}$ and

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

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

Step 7.2

Multiply

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

$\frac{2}{3}$ .

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

Step 7.3

Multiply.

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Step 7.3.1

Multiply

$2$ by

$3$ .

$\frac{6 {(x^{2} + 1)}^{\frac{1}{2}}}{2 \cdot 3} \frac{d}{d x} [x^{2} + 1]$

Step 7.3.2

Multiply

$2$ by

$3$ .

$\frac{6 {(x^{2} + 1)}^{\frac{1}{2}}}{6} \frac{d}{d x} [x^{2} + 1]$

Step 7.4

Cancel the common factor.

$\frac{6 {(x^{2} + 1)}^{\frac{1}{2}}}{6} \frac{d}{d x} [x^{2} + 1]$

Step 7.5

Divide

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

$1$ .

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

Step 8

By the Sum Rule, the derivative of

$x^{2} + 1$ with respect to

$x$ is

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

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

Step 9

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

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

Step 10

Since

$1$ is constant with respect to

$x$ , the derivative of

$1$ with respect to

$x$ is

$0$ .

${(x^{2} + 1)}^{\frac{1}{2}} (2 x + 0)$

Step 11

Simplify the expression.

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Step 11.1

Add

$2 x$ and

$0$ .

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

Step 11.2

Move

$2$ to the left of

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

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

Step 11.3

Reorder the factors of

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

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