Calculus Examples

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

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $\frac{x}{1 + x^{2}}$ .

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

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{x}{1 + x^{2}}$ .

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

Step 3

Multiply $\frac{1 + x^{2}}{x}$ by $1$ .

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

Step 4

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = 1 + x^{2}$ .

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

Step 5

Differentiate.

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Step 5.1

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

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

Step 5.2

Multiply $1 + x^{2}$ by $1$ .

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

Step 5.3

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

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

Step 5.4

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

$\frac{1 + x^{2}}{x} \cdot \frac{1 + x^{2} - x (0 + \frac{d}{d x} [x^{2}])}{{(1 + x^{2})}^{2}}$

Step 5.5

Add $0$ and $\frac{d}{d x} [x^{2}]$ .

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

Step 5.6

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

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

Step 5.7

Multiply $2$ by $- 1$ .

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

Step 6

Raise $x$ to the power of $1$ .

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

Step 7

Raise $x$ to the power of $1$ .

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

Step 8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 9

Add $1$ and $1$ .

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

Step 10

Subtract $2 x^{2}$ from $x^{2}$ .

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

Step 11

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

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

Step 12

Cancel the common factors.

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Step 12.1

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

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

Step 12.2

Cancel the common factor.

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

Step 12.3

Rewrite the expression.

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

Step 13

Simplify.

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Step 13.1

Apply the distributive property.

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

Step 13.2

Combine terms.

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Step 13.2.1

Multiply $x$ by $1$ .

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

Step 13.2.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 13.2.2.1

Multiply $x$ by $x^{2}$ .

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Step 13.2.2.1.1

Raise $x$ to the power of $1$ .

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

Step 13.2.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 13.2.2.2

Add $1$ and $2$ .

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

Step 13.3

Reorder terms.

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