Calculus Examples

$\frac{- 6}{{(x - 1)}^{2}}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{6}{{(x - 1)}^{2}}]$

Step 1.2

Since $- 6$ is constant with respect to $x$ , the derivative of $- \frac{6}{{(x - 1)}^{2}}$ with respect to $x$ is $- 6 \frac{d}{d x} [\frac{1}{{(x - 1)}^{2}}]$ .

$- 6 \frac{d}{d x} [\frac{1}{{(x - 1)}^{2}}]$

Step 1.3

Apply basic rules of exponents.

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Step 1.3.1

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

$- 6 \frac{d}{d x} [{({(x - 1)}^{2})}^{- 1}]$

Step 1.3.2

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

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Step 1.3.2.1

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

$- 6 \frac{d}{d x} [{(x - 1)}^{2 \cdot - 1}]$

Step 1.3.2.2

Multiply $2$ by $- 1$ .

$- 6 \frac{d}{d x} [{(x - 1)}^{- 2}]$

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

$- 6 (\frac{d}{d u} [u^{- 2}] \frac{d}{d x} [x - 1])$

Step 2.2

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

$- 6 (- 2 u^{- 3} \frac{d}{d x} [x - 1])$

Step 2.3

Replace all occurrences of $u$ with $x - 1$ .

$- 6 (- 2 {(x - 1)}^{- 3} \frac{d}{d x} [x - 1])$

Step 3

Differentiate.

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Step 3.1

Multiply $- 2$ by $- 6$ .

$12 ({(x - 1)}^{- 3} \frac{d}{d x} [x - 1])$

Step 3.2

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

$12 {(x - 1)}^{- 3} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 3.3

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

$12 {(x - 1)}^{- 3} (1 + \frac{d}{d x} [- 1])$

Step 3.4

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

$12 {(x - 1)}^{- 3} (1 + 0)$

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $1$ and $0$ .

$12 {(x - 1)}^{- 3} \cdot 1$

Step 3.5.2

Multiply $12$ by $1$ .

$12 {(x - 1)}^{- 3}$

Step 4

Simplify.

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Step 4.1

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

$12 \frac{1}{{(x - 1)}^{3}}$

Step 4.2

Combine $12$ and $\frac{1}{{(x - 1)}^{3}}$ .

$\frac{12}{{(x - 1)}^{3}}$