Calculus Examples

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

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Differentiate using the Power Rule.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Combine fractions.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Combine terms.

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Step 3.3.1

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

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

Step 3.3.2

One to any power is one.

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

Step 3.3.3

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

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

Step 3.3.4

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

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Step 3.3.4.1

Cancel the common factor.

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

Step 3.3.4.2

Rewrite the expression.

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