Calculus Examples

$f (x) = \frac{8}{\sqrt{4 - x}}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

$\frac{d}{d x} [\frac{8}{{(4 - x)}^{\frac{1}{2}}}]$

Step 1.2

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8}{{(4 - x)}^{\frac{1}{2}}}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{1}{{(4 - x)}^{\frac{1}{2}}}]$ .

$8 \frac{d}{d x} [\frac{1}{{(4 - x)}^{\frac{1}{2}}}]$

Step 1.3

Apply basic rules of exponents.

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Step 1.3.1

Rewrite $\frac{1}{{(4 - x)}^{\frac{1}{2}}}$ as ${({(4 - x)}^{\frac{1}{2}})}^{- 1}$ .

$8 \frac{d}{d x} [{({(4 - x)}^{\frac{1}{2}})}^{- 1}]$

Step 1.3.2

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

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Step 1.3.2.1

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

$8 \frac{d}{d x} [{(4 - x)}^{\frac{1}{2} \cdot - 1}]$

Step 1.3.2.2

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

$8 \frac{d}{d x} [{(4 - x)}^{\frac{- 1}{2}}]$

Step 1.3.2.3

Move the negative in front of the fraction.

$8 \frac{d}{d x} [{(4 - x)}^{- \frac{1}{2}}]$

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 $g (x) = 4 - x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $4 - x$ .

$8 (\frac{d}{d u} [u^{- \frac{1}{2}}] \frac{d}{d x} [4 - x])$

Step 2.2

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

$8 (- \frac{1}{2} u^{- \frac{1}{2} - 1} \frac{d}{d x} [4 - x])$

Step 2.3

Replace all occurrences of $u$ with $4 - x$ .

$8 (- \frac{1}{2} {(4 - x)}^{- \frac{1}{2} - 1} \frac{d}{d x} [4 - x])$

Step 3

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

$8 (- \frac{1}{2} {(4 - x)}^{- \frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [4 - x])$

Step 4

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

$8 (- \frac{1}{2} {(4 - x)}^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [4 - x])$

Step 5

Combine the numerators over the common denominator.

$8 (- \frac{1}{2} {(4 - x)}^{\frac{- 1 - 1 \cdot 2}{2}} \frac{d}{d x} [4 - x])$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$8 (- \frac{1}{2} {(4 - x)}^{\frac{- 1 - 2}{2}} \frac{d}{d x} [4 - x])$

Step 6.2

Subtract $2$ from $- 1$ .

$8 (- \frac{1}{2} {(4 - x)}^{\frac{- 3}{2}} \frac{d}{d x} [4 - x])$

Step 7

Move the negative in front of the fraction.

$8 (- \frac{1}{2} {(4 - x)}^{- \frac{3}{2}} \frac{d}{d x} [4 - x])$

Step 8

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

$8 (- \frac{{(4 - x)}^{- \frac{3}{2}}}{2} \frac{d}{d x} [4 - x])$

Step 9

Simplify the expression.

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Step 9.1

Move ${(4 - x)}^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$8 (- \frac{1}{2 {(4 - x)}^{\frac{3}{2}}} \frac{d}{d x} [4 - x])$

Step 9.2

Multiply $- 1$ by $8$ .

$- 8 (\frac{1}{2 {(4 - x)}^{\frac{3}{2}}} \frac{d}{d x} [4 - x])$

Step 10

Combine $\frac{1}{2 {(4 - x)}^{\frac{3}{2}}}$ and $- 8$ .

$\frac{- 8}{2 {(4 - x)}^{\frac{3}{2}}} \frac{d}{d x} [4 - x]$

Step 11

Factor $2$ out of $- 8$ .

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

Step 12

Cancel the common factors.

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Step 12.1

Factor $2$ out of $2 {(4 - x)}^{\frac{3}{2}}$ .

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

Step 12.2

Cancel the common factor.

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

Step 12.3

Rewrite the expression.

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

Step 13

Move the negative in front of the fraction.

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

Step 14

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

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

Step 15

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

$- \frac{4}{{(4 - x)}^{\frac{3}{2}}} (0 + \frac{d}{d x} [- x])$

Step 16

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 17

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

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

Step 18

Multiply.

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Step 18.1

Multiply $- 1$ by $- 1$ .

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

Step 18.2

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

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

Step 19

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

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

Step 20

Simplify the expression.

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Step 20.1

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

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

Step 20.2

Reorder terms.

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