Calculus Examples

$y = \arctan (\sqrt{x - 4})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 4}$ as ${(x - 4)}^{\frac{1}{2}}$ .

$y = \arctan ({(x - 4)}^{\frac{1}{2}})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\arctan ({(x - 4)}^{\frac{1}{2}}))$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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 $g (x) = {(x - 4)}^{\frac{1}{2}}$ .

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Step 4.1.1

To apply the Chain Rule, set $u_{1}$ as ${(x - 4)}^{\frac{1}{2}}$ .

$\frac{d}{d u_{1}} [\arctan (u_{1})] \frac{d}{d x} [{(x - 4)}^{\frac{1}{2}}]$

Step 4.1.2

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

$\frac{1}{1 + {u_{1}}^{2}} \frac{d}{d x} [{(x - 4)}^{\frac{1}{2}}]$

Step 4.1.3

Replace all occurrences of $u_{1}$ with ${(x - 4)}^{\frac{1}{2}}$ .

$\frac{1}{1 + {({(x - 4)}^{\frac{1}{2}})}^{2}} \frac{d}{d x} [{(x - 4)}^{\frac{1}{2}}]$

Step 4.2

Multiply the exponents in ${({(x - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1

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

$\frac{1}{1 + {(x - 4)}^{\frac{1}{2} \cdot 2}} \frac{d}{d x} [{(x - 4)}^{\frac{1}{2}}]$

Step 4.2.2

Cancel the common factor of $2$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{1}{1 + {(x - 4)}^{\frac{1}{2} \cdot 2}} \frac{d}{d x} [{(x - 4)}^{\frac{1}{2}}]$

Step 4.2.2.2

Rewrite the expression.

$\frac{1}{1 + {(x - 4)}^{1}} \frac{d}{d x} [{(x - 4)}^{\frac{1}{2}}]$

Step 4.3

Simplify.

$\frac{1}{1 + x - 4} \frac{d}{d x} [{(x - 4)}^{\frac{1}{2}}]$

Step 4.4

Subtract $4$ from $1$ .

$\frac{1}{x - 3} \frac{d}{d x} [{(x - 4)}^{\frac{1}{2}}]$

Step 4.5

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 $g (x) = x - 4$ .

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Step 4.5.1

To apply the Chain Rule, set $u_{2}$ as $x - 4$ .

$\frac{1}{x - 3} (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [x - 4])$

Step 4.5.2

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

$\frac{1}{x - 3} (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [x - 4])$

Step 4.5.3

Replace all occurrences of $u_{2}$ with $x - 4$ .

$\frac{1}{x - 3} (\frac{1}{2} {(x - 4)}^{\frac{1}{2} - 1} \frac{d}{d x} [x - 4])$

Step 4.6

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

$\frac{1}{x - 3} (\frac{1}{2} {(x - 4)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x - 4])$

Step 4.7

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

$\frac{1}{x - 3} (\frac{1}{2} {(x - 4)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x - 4])$

Step 4.8

Combine the numerators over the common denominator.

$\frac{1}{x - 3} (\frac{1}{2} {(x - 4)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x - 4])$

Step 4.9

Simplify the numerator.

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Step 4.9.1

Multiply $- 1$ by $2$ .

$\frac{1}{x - 3} (\frac{1}{2} {(x - 4)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x - 4])$

Step 4.9.2

Subtract $2$ from $1$ .

$\frac{1}{x - 3} (\frac{1}{2} {(x - 4)}^{\frac{- 1}{2}} \frac{d}{d x} [x - 4])$

Step 4.10

Combine fractions.

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Step 4.10.1

Move the negative in front of the fraction.

$\frac{1}{x - 3} (\frac{1}{2} {(x - 4)}^{- \frac{1}{2}} \frac{d}{d x} [x - 4])$

Step 4.10.2

Combine $\frac{1}{2}$ and ${(x - 4)}^{- \frac{1}{2}}$ .

$\frac{1}{x - 3} (\frac{{(x - 4)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x - 4])$

Step 4.10.3

Move ${(x - 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x - 3} (\frac{1}{2 {(x - 4)}^{\frac{1}{2}}} \frac{d}{d x} [x - 4])$

Step 4.10.4

Multiply $\frac{1}{2 {(x - 4)}^{\frac{1}{2}}}$ by $\frac{1}{x - 3}$ .

$\frac{1}{2 {(x - 4)}^{\frac{1}{2}} (x - 3)} \frac{d}{d x} [x - 4]$

Step 4.11

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$\frac{1}{2 {(x - 4)}^{\frac{1}{2}} (x - 3)} (\frac{d}{d x} [x] + \frac{d}{d x} [- 4])$

Step 4.12

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

$\frac{1}{2 {(x - 4)}^{\frac{1}{2}} (x - 3)} (1 + \frac{d}{d x} [- 4])$

Step 4.13

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{1}{2 {(x - 4)}^{\frac{1}{2}} (x - 3)} (1 + 0)$

Step 4.14

Simplify the expression.

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Step 4.14.1

Add $1$ and $0$ .

$\frac{1}{2 {(x - 4)}^{\frac{1}{2}} (x - 3)} \cdot 1$

Step 4.14.2

Multiply $\frac{1}{2 {(x - 4)}^{\frac{1}{2}} (x - 3)}$ by $1$ .

$\frac{1}{2 {(x - 4)}^{\frac{1}{2}} (x - 3)}$

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = \frac{1}{2 {(x - 4)}^{\frac{1}{2}} (x - 3)}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{1}{2 {(x - 4)}^{\frac{1}{2}} (x - 3)}$