Calculus Examples

$y = \ln ({(x^{2} + 1)}^{3})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = {(x^{2} + 1)}^{3}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as ${(x^{2} + 1)}^{3}$ .

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

Step 1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

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

Step 1.3

Replace all occurrences of $u_{1}$ with ${(x^{2} + 1)}^{3}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x^{2} + 1$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $x^{2} + 1$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{2}$ with $x^{2} + 1$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Simplify terms.

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Step 3.2.1

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

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

Step 3.2.2

Move $3$ to the left of ${(x^{2} + 1)}^{2}$ .

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

Step 3.2.3

Cancel the common factor of ${(x^{2} + 1)}^{2}$ and ${(x^{2} + 1)}^{3}$ .

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Step 3.2.3.1

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

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

Step 3.2.3.2

Cancel the common factors.

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Step 3.2.3.2.1

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

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

Step 3.2.3.2.2

Cancel the common factor.

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

Step 3.2.3.2.3

Rewrite the expression.

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

Step 3.3

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Combine fractions.

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Step 3.6.1

Add $2 x$ and $0$ .

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

Step 3.6.2

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

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

Step 3.6.3

Multiply $2$ by $3$ .

$\frac{6}{x^{2} + 1} x$

Step 3.6.4

Combine $\frac{6}{x^{2} + 1}$ and $x$ .

$\frac{6 x}{x^{2} + 1}$