Trigonometry Examples

$(\arctan (\frac{5}{x}))$

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 $g (x) = \frac{5}{x}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{5}{x}$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [\frac{5}{x}]$

Step 1.2

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

$\frac{1}{1 + u^{2}} \frac{d}{d x} [\frac{5}{x}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{5}{x}$ .

$\frac{1}{1 + {(\frac{5}{x})}^{2}} \frac{d}{d x} [\frac{5}{x}]$

Step 2

Differentiate.

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{x}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{1}{1 + {(\frac{5}{x})}^{2}} (5 \frac{d}{d x} [\frac{1}{x}])$

Step 2.2

Combine fractions.

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Step 2.2.1

Combine $5$ and $\frac{1}{1 + {(\frac{5}{x})}^{2}}$ .

$\frac{5}{1 + {(\frac{5}{x})}^{2}} \frac{d}{d x} [\frac{1}{x}]$

Step 2.2.2

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

$\frac{5}{1 + {(\frac{5}{x})}^{2}} \frac{d}{d x} [x^{- 1}]$

Step 2.3

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

$\frac{5}{1 + {(\frac{5}{x})}^{2}} (- x^{- 2})$

Step 2.4

Combine fractions.

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Step 2.4.1

Combine $\frac{5}{1 + {(\frac{5}{x})}^{2}}$ and $x^{- 2}$ .

$- \frac{5 x^{- 2}}{1 + {(\frac{5}{x})}^{2}}$

Step 2.4.2

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

$- \frac{5}{(1 + {(\frac{5}{x})}^{2}) x^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the product rule to $\frac{5}{x}$ .

$- \frac{5}{(1 + \frac{5^{2}}{x^{2}}) x^{2}}$

Step 3.2

Apply the distributive property.

$- \frac{5}{1 x^{2} + \frac{5^{2}}{x^{2}} x^{2}}$

Step 3.3

Combine terms.

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Step 3.3.1

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

$- \frac{5}{x^{2} + \frac{5^{2}}{x^{2}} x^{2}}$

Step 3.3.2

Raise $5$ to the power of $2$ .

$- \frac{5}{x^{2} + \frac{25}{x^{2}} x^{2}}$

Step 3.3.3

Combine $\frac{25}{x^{2}}$ and $x^{2}$ .

$- \frac{5}{x^{2} + \frac{25 x^{2}}{x^{2}}}$

Step 3.3.4

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

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Step 3.3.4.1

Cancel the common factor.

$- \frac{5}{x^{2} + \frac{25 x^{2}}{x^{2}}}$

Step 3.3.4.2

Divide $25$ by $1$ .

$- \frac{5}{x^{2} + 25}$