Calculus Examples

$\frac{2}{{(2 - 11 x)}^{2}}$

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 $\frac{2}{{(2 - 11 x)}^{2}}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{{(2 - 11 x)}^{2}}]$ .

$2 \frac{d}{d x} [\frac{1}{{(2 - 11 x)}^{2}}]$

Step 1.2

Apply basic rules of exponents.

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Step 1.2.1

Rewrite $\frac{1}{{(2 - 11 x)}^{2}}$ as ${({(2 - 11 x)}^{2})}^{- 1}$ .

$2 \frac{d}{d x} [{({(2 - 11 x)}^{2})}^{- 1}]$

Step 1.2.2

Multiply the exponents in ${({(2 - 11 x)}^{2})}^{- 1}$ .

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Step 1.2.2.1

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

$2 \frac{d}{d x} [{(2 - 11 x)}^{2 \cdot - 1}]$

Step 1.2.2.2

Multiply $2$ by $- 1$ .

$2 \frac{d}{d x} [{(2 - 11 x)}^{- 2}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 2}$ and $g (x) = 2 - 11 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $2 - 11 x$ .

$2 (\frac{d}{d u} [u^{- 2}] \frac{d}{d x} [2 - 11 x])$

Step 2.2

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

$2 (- 2 u^{- 3} \frac{d}{d x} [2 - 11 x])$

Step 2.3

Replace all occurrences of $u$ with $2 - 11 x$ .

$2 (- 2 {(2 - 11 x)}^{- 3} \frac{d}{d x} [2 - 11 x])$

Step 3

Differentiate.

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Step 3.1

Multiply $- 2$ by $2$ .

$- 4 ({(2 - 11 x)}^{- 3} \frac{d}{d x} [2 - 11 x])$

Step 3.2

By the Sum Rule, the derivative of $2 - 11 x$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- 11 x]$ .

$- 4 {(2 - 11 x)}^{- 3} (\frac{d}{d x} [2] + \frac{d}{d x} [- 11 x])$

Step 3.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$- 4 {(2 - 11 x)}^{- 3} (0 + \frac{d}{d x} [- 11 x])$

Step 3.4

Add $0$ and $\frac{d}{d x} [- 11 x]$ .

$- 4 {(2 - 11 x)}^{- 3} \frac{d}{d x} [- 11 x]$

Step 3.5

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11 x$ with respect to $x$ is $- 11 \frac{d}{d x} [x]$ .

$- 4 {(2 - 11 x)}^{- 3} (- 11 \frac{d}{d x} [x])$

Step 3.6

Multiply $- 11$ by $- 4$ .

$44 {(2 - 11 x)}^{- 3} \frac{d}{d x} [x]$

Step 3.7

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

$44 {(2 - 11 x)}^{- 3} \cdot 1$

Step 3.8

Multiply $44$ by $1$ .

$44 {(2 - 11 x)}^{- 3}$

Step 4

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

$44 \frac{1}{{(2 - 11 x)}^{3}}$

Step 5

Simplify.

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Step 5.1

Combine $44$ and $\frac{1}{{(2 - 11 x)}^{3}}$ .

$\frac{44}{{(2 - 11 x)}^{3}}$

Step 5.2

Reorder terms.

$\frac{44}{{(- 11 x + 2)}^{3}}$