Calculus Examples

$y = \sqrt{1 + \cot^{2} (x)}$

Step 1

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

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

Step 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 $g (x) = 1 + \cot^{2} (x)$ .

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

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

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

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

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Step 7

Differentiate.

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Move the negative in front of the fraction.

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

Combine fractions.

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

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

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

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

By the Sum Rule, the derivative of $1 + \cot^{2} (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\cot^{2} (x)]$ .

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

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

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

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

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

Step 8

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 $\cot (x)$ .

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

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

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

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

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

Step 9

Simplify terms.

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

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 10

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

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

Step 11

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

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

Step 12

Simplify.

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Apply pythagorean identity.

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

Simplify the denominator.

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Multiply the exponents in ${(\csc^{2} (x))}^{\frac{1}{2}}$ .

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Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Cancel the common factor of $\csc^{2} (x)$ and $\csc (x)$ .

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Factor $\csc (x)$ out of $\cot (x) \csc^{2} (x)$ .

$- \frac{\csc (x) (\cot (x) \csc (x))}{\csc (x)}$

Cancel the common factors.

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Multiply by $1$ .

$- \frac{\csc (x) (\cot (x) \csc (x))}{\csc (x) \cdot 1}$

Cancel the common factor.

$- \frac{\csc (x) (\cot (x) \csc (x))}{\csc (x) \cdot 1}$

Rewrite the expression.

$- \frac{\cot (x) \csc (x)}{1}$

Divide $\cot (x) \csc (x)$ by $1$ .

$- (\cot (x) \csc (x))$

$- \cot (x) \csc (x)$