Calculus Examples

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

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

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

Step 3.4.2

Multiply $x + 1$ by $1$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Combine fractions.

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Step 3.8.1

Add $1$ and $0$ .

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

Step 3.8.2

Multiply $- 1$ by $1$ .

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

Step 3.8.3

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Simplify the numerator.

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Step 4.3.1

Combine the opposite terms in $x + 1 - x - - 1$ .

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Step 4.3.1.1

Subtract $x$ from $x$ .

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

Step 4.3.1.2

Add $0$ and $1$ .

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

Step 4.3.2

Multiply $- 1$ by $- 1$ .

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

Step 4.3.3

Add $1$ and $1$ .

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

Step 4.4

Combine terms.

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Step 4.4.1

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

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

Step 4.4.2

Combine the numerators over the common denominator.

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

Step 4.4.3

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

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

Step 4.4.4

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

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Step 4.4.4.1

Cancel the common factor.

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

Step 4.4.4.2

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

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

Step 4.5

Simplify the denominator.

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Step 4.5.1

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

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

Step 4.5.2

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

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Step 4.5.2.1

Apply the distributive property.

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

Step 4.5.2.2

Apply the distributive property.

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

Step 4.5.2.3

Apply the distributive property.

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

Step 4.5.3

Simplify and combine like terms.

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Step 4.5.3.1

Simplify each term.

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Step 4.5.3.1.1

Multiply $x$ by $x$ .

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

Step 4.5.3.1.2

Multiply $x$ by $1$ .

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

Step 4.5.3.1.3

Multiply $x$ by $1$ .

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

Step 4.5.3.1.4

Multiply $1$ by $1$ .

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

Step 4.5.3.2

Add $x$ and $x$ .

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

Step 4.5.4

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

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

Step 4.5.5

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

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Step 4.5.5.1

Apply the distributive property.

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

Step 4.5.5.2

Apply the distributive property.

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

Step 4.5.5.3

Apply the distributive property.

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

Step 4.5.6

Simplify and combine like terms.

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Step 4.5.6.1

Simplify each term.

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Step 4.5.6.1.1

Multiply $x$ by $x$ .

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

Step 4.5.6.1.2

Move $- 1$ to the left of $x$ .

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

Step 4.5.6.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 4.5.6.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 4.5.6.1.5

Multiply $- 1$ by $- 1$ .

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

Step 4.5.6.2

Subtract $x$ from $- x$ .

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

Step 4.5.7

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

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

Step 4.5.8

Subtract $2 x$ from $2 x$ .

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

Step 4.5.9

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

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

Step 4.5.10

Add $1$ and $1$ .

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

Step 4.5.11

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

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Step 4.5.11.1

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

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

Step 4.5.11.2

Factor $2$ out of $2$ .

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

Step 4.5.11.3

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

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

Step 4.6

Cancel the common factor of $2$ .

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Step 4.6.1

Cancel the common factor.

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

Step 4.6.2

Rewrite the expression.

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