Calculus Examples

$\arctan (\frac{1 + x}{1 - x})$

Step 1

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

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Step 1.1

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

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [\frac{1 + x}{1 - x}]$

Step 1.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

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

Step 1.3

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

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

Step 2

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 $f (x) = 1 + x$ and $g (x) = 1 - x$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Multiply $1 - x$ by $1$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 3.10

Multiply.

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Step 3.10.1

Multiply $- 1$ by $- 1$ .

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

Step 3.10.2

Multiply $1 + x$ by $1$ .

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

Step 3.11

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

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

Step 3.12

Simplify terms.

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Step 3.12.1

Multiply $1 + x$ by $1$ .

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

Step 3.12.2

Add $1$ and $1$ .

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

Step 3.12.3

Add $- x$ and $x$ .

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

Step 3.12.4

Add $0$ and $2$ .

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

Step 3.12.5

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

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

Step 4

Simplify.

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Step 4.1

Apply the product rule to $\frac{1 + x}{1 - x}$ .

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

Step 4.2

Combine terms.

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Step 4.2.1

Write $1$ as a fraction with a common denominator.

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

Step 4.2.2

Combine the numerators over the common denominator.

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

Step 4.2.3

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

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

Step 4.2.4

Cancel the common factor of ${(1 - x)}^{2}$ .

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Step 4.2.4.1

Cancel the common factor.

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

Step 4.2.4.2

Divide ${(1 - x)}^{2} + {(1 + x)}^{2}$ by $1$ .

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

Step 4.3

Simplify the denominator.

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Step 4.3.1

Rewrite ${(1 - x)}^{2}$ as $(1 - x) (1 - x)$ .

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

Step 4.3.2

Expand $(1 - x) (1 - x)$ using the FOIL Method.

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Step 4.3.2.1

Apply the distributive property.

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

Step 4.3.2.2

Apply the distributive property.

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

Step 4.3.2.3

Apply the distributive property.

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

Step 4.3.3

Simplify and combine like terms.

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Step 4.3.3.1

Simplify each term.

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Step 4.3.3.1.1

Multiply $1$ by $1$ .

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

Step 4.3.3.1.2

Multiply $- x$ by $1$ .

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

Step 4.3.3.1.3

Multiply $- 1$ by $1$ .

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

Step 4.3.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.3.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.3.1.5.1

Move $x$ .

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

Step 4.3.3.1.5.2

Multiply $x$ by $x$ .

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

Step 4.3.3.1.6

Multiply $- 1$ by $- 1$ .

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

Step 4.3.3.1.7

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

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

Step 4.3.3.2

Subtract $x$ from $- x$ .

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

Step 4.3.4

Rewrite ${(1 + x)}^{2}$ as $(1 + x) (1 + x)$ .

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

Step 4.3.5

Expand $(1 + x) (1 + x)$ using the FOIL Method.

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Step 4.3.5.1

Apply the distributive property.

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

Step 4.3.5.2

Apply the distributive property.

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

Step 4.3.5.3

Apply the distributive property.

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

Step 4.3.6

Simplify and combine like terms.

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Step 4.3.6.1

Simplify each term.

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Step 4.3.6.1.1

Multiply $1$ by $1$ .

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

Step 4.3.6.1.2

Multiply $x$ by $1$ .

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

Step 4.3.6.1.3

Multiply $x$ by $1$ .

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

Step 4.3.6.1.4

Multiply $x$ by $x$ .

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

Step 4.3.6.2

Add $x$ and $x$ .

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

Step 4.3.7

Add $1$ and $1$ .

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

Step 4.3.8

Add $- 2 x$ and $2 x$ .

$\frac{2}{x^{2} + 2 + 0 + x^{2}}$

Step 4.3.9

Add $x^{2}$ and $0$ .

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

Step 4.3.10

Add $x^{2}$ and $x^{2}$ .

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

Step 4.3.11

Factor $2$ out of $2 x^{2} + 2$ .

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Step 4.3.11.1

Factor $2$ out of $2 x^{2}$ .

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

Step 4.3.11.2

Factor $2$ out of $2$ .

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

Step 4.3.11.3

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

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

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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