Calculus Examples

$\sqrt{9 + \cot^{2} (x)}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 + \cot^{2} (x)}$ as ${(9 + \cot^{2} (x))}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(9 + \cot^{2} (x))}^{\frac{1}{2}}]$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 9 + \cot^{2} (x)$ .

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To apply the Chain Rule, set $u_{1}$ as $9 + \cot^{2} (x)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [9 + \cot^{2} (x)]$

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [9 + \cot^{2} (x)]$

Replace all occurrences of $u_{1}$ with $9 + \cot^{2} (x)$ .

$\frac{1}{2} {(9 + \cot^{2} (x))}^{\frac{1}{2} - 1} \frac{d}{d x} [9 + \cot^{2} (x)]$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} {(9 + \cot^{2} (x))}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [9 + \cot^{2} (x)]$

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} {(9 + \cot^{2} (x))}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [9 + \cot^{2} (x)]$

Combine the numerators over the common denominator.

$\frac{1}{2} {(9 + \cot^{2} (x))}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [9 + \cot^{2} (x)]$

Simplify the numerator.

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Multiply $- 1$ by $2$ .

$\frac{1}{2} {(9 + \cot^{2} (x))}^{\frac{1 - 2}{2}} \frac{d}{d x} [9 + \cot^{2} (x)]$

Subtract $2$ from $1$ .

$\frac{1}{2} {(9 + \cot^{2} (x))}^{\frac{- 1}{2}} \frac{d}{d x} [9 + \cot^{2} (x)]$

Differentiate.

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Move the negative in front of the fraction.

$\frac{1}{2} {(9 + \cot^{2} (x))}^{- \frac{1}{2}} \frac{d}{d x} [9 + \cot^{2} (x)]$

Combine fractions.

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Combine $\frac{1}{2}$ and ${(9 + \cot^{2} (x))}^{- \frac{1}{2}}$ .

$\frac{{(9 + \cot^{2} (x))}^{- \frac{1}{2}}}{2} \frac{d}{d x} [9 + \cot^{2} (x)]$

Move ${(9 + \cot^{2} (x))}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(9 + \cot^{2} (x))}^{\frac{1}{2}}} \frac{d}{d x} [9 + \cot^{2} (x)]$

By the Sum Rule, the derivative of $9 + \cot^{2} (x)$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [\cot^{2} (x)]$ .

$\frac{1}{2 {(9 + \cot^{2} (x))}^{\frac{1}{2}}} (\frac{d}{d x} [9] + \frac{d}{d x} [\cot^{2} (x)])$

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{1}{2 {(9 + \cot^{2} (x))}^{\frac{1}{2}}} (0 + \frac{d}{d x} [\cot^{2} (x)])$

Add $0$ and $\frac{d}{d x} [\cot^{2} (x)]$ .

$\frac{1}{2 {(9 + \cot^{2} (x))}^{\frac{1}{2}}} \frac{d}{d x} [\cot^{2} (x)]$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cot (x)$ .

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To apply the Chain Rule, set $u_{2}$ as $\cot (x)$ .

$\frac{1}{2 {(9 + \cot^{2} (x))}^{\frac{1}{2}}} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\cot (x)])$

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$\frac{1}{2 {(9 + \cot^{2} (x))}^{\frac{1}{2}}} (2 u_{2} \frac{d}{d x} [\cot (x)])$

Replace all occurrences of $u_{2}$ with $\cot (x)$ .

$\frac{1}{2 {(9 + \cot^{2} (x))}^{\frac{1}{2}}} (2 \cot (x) \frac{d}{d x} [\cot (x)])$

Simplify terms.

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Combine $2$ and $\frac{1}{2 {(9 + \cot^{2} (x))}^{\frac{1}{2}}}$ .

$\frac{2}{2 {(9 + \cot^{2} (x))}^{\frac{1}{2}}} (\cot (x) \frac{d}{d x} [\cot (x)])$

Combine $\cot (x)$ and $\frac{2}{2 {(9 + \cot^{2} (x))}^{\frac{1}{2}}}$ .

$\frac{\cot (x) \cdot 2}{2 {(9 + \cot^{2} (x))}^{\frac{1}{2}}} \frac{d}{d x} [\cot (x)]$

Move $2$ to the left of $\cot (x)$ .

$\frac{2 \cdot \cot (x)}{2 {(9 + \cot^{2} (x))}^{\frac{1}{2}}} \frac{d}{d x} [\cot (x)]$

Cancel the common factor.

$\frac{2 \cot (x)}{2 {(9 + \cot^{2} (x))}^{\frac{1}{2}}} \frac{d}{d x} [\cot (x)]$

Rewrite the expression.

$\frac{\cot (x)}{{(9 + \cot^{2} (x))}^{\frac{1}{2}}} \frac{d}{d x} [\cot (x)]$

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$\frac{\cot (x)}{{(9 + \cot^{2} (x))}^{\frac{1}{2}}} (- \csc^{2} (x))$

Combine $\frac{\cot (x)}{{(9 + \cot^{2} (x))}^{\frac{1}{2}}}$ and $\csc^{2} (x)$ .

$- \frac{\cot (x) \csc^{2} (x)}{{(9 + \cot^{2} (x))}^{\frac{1}{2}}}$