Calculus Examples

$y = {(\frac{x}{2} - 1)}^{- 10}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 10}$ and $g (x) = \frac{x}{2} - 1$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{x}{2} - 1$ .

$\frac{d}{d u} [u^{- 10}] \frac{d}{d x} [\frac{x}{2} - 1]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 10$ .

$- 10 u^{- 11} \frac{d}{d x} [\frac{x}{2} - 1]$

Step 1.3

Replace all occurrences of $u$ with $\frac{x}{2} - 1$ .

$- 10 {(\frac{x}{2} - 1)}^{- 11} \frac{d}{d x} [\frac{x}{2} - 1]$

Step 2

Differentiate.

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Step 2.1

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

$- 10 {(\frac{x}{2} - 1)}^{- 11} (\frac{d}{d x} [\frac{x}{2}] + \frac{d}{d x} [- 1])$

Step 2.2

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

$- 10 {(\frac{x}{2} - 1)}^{- 11} (\frac{1}{2} \frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 2.3

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

$- 10 {(\frac{x}{2} - 1)}^{- 11} (\frac{1}{2} \cdot 1 + \frac{d}{d x} [- 1])$

Step 2.4

Multiply $\frac{1}{2}$ by $1$ .

$- 10 {(\frac{x}{2} - 1)}^{- 11} (\frac{1}{2} + \frac{d}{d x} [- 1])$

Step 2.5

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

$- 10 {(\frac{x}{2} - 1)}^{- 11} (\frac{1}{2} + 0)$

Step 2.6

Simplify terms.

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Step 2.6.1

Add $\frac{1}{2}$ and $0$ .

$- 10 {(\frac{x}{2} - 1)}^{- 11} \frac{1}{2}$

Step 2.6.2

Combine $\frac{1}{2}$ and $- 10$ .

$\frac{- 10}{2} {(\frac{x}{2} - 1)}^{- 11}$

Step 2.6.3

Combine $\frac{- 10}{2}$ and ${(\frac{x}{2} - 1)}^{- 11}$ .

$\frac{- 10 {(\frac{x}{2} - 1)}^{- 11}}{2}$

Step 2.6.4

Move ${(\frac{x}{2} - 1)}^{- 11}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 10}{2 {(\frac{x}{2} - 1)}^{11}}$

Step 2.6.5

Cancel the common factor of $- 10$ and $2$ .

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Step 2.6.5.1

Factor $2$ out of $- 10$ .

$\frac{2 \cdot - 5}{2 {(\frac{x}{2} - 1)}^{11}}$

Step 2.6.5.2

Cancel the common factors.

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Step 2.6.5.2.1

Factor $2$ out of $2 {(\frac{x}{2} - 1)}^{11}$ .

$\frac{2 \cdot - 5}{2 ({(\frac{x}{2} - 1)}^{11})}$

Step 2.6.5.2.2

Cancel the common factor.

$\frac{2 \cdot - 5}{2 {(\frac{x}{2} - 1)}^{11}}$

Step 2.6.5.2.3

Rewrite the expression.

$\frac{- 5}{{(\frac{x}{2} - 1)}^{11}}$

Step 2.6.6

Move the negative in front of the fraction.

$- \frac{5}{{(\frac{x}{2} - 1)}^{11}}$