Calculus Examples

$\frac{2}{4 - x}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

$2 \frac{d}{d x} [\frac{1}{4 - x}]$

Step 1.2

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 4 - x$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $2$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Multiply $- 1$ by $- 2$ .

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

Step 3.7

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

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

Step 3.8

Multiply $2$ by $1$ .

$2 {(4 - x)}^{- 2}$

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Reorder terms.

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