Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Evaluate $\frac{d}{d x} [\arctan (\frac{1}{x})]$ .

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Combine terms.

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Step 4.3.1

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

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

Step 4.3.2

One to any power is one.

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

Step 4.3.3

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

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

Step 4.3.4

Cancel the common factor of $x^{2}$ .

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Step 4.3.4.1

Cancel the common factor.

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

Step 4.3.4.2

Rewrite the expression.

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

Step 4.3.5

Reorder terms.

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

Step 4.3.6

Combine the numerators over the common denominator.

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

Step 4.3.7

Subtract $1$ from $1$ .

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

Step 4.4

Divide $0$ by $x^{2} + 1$ .

$0$