Calculus Examples

${(x^{2} - 7 x - 1)}^{- 8}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 8}$ and $g (x) = x^{2} - 7 x - 1$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{2} - 7 x - 1$ .

$\frac{d}{d u} [u^{- 8}] \frac{d}{d x} [x^{2} - 7 x - 1]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 8$ .

$- 8 u^{- 9} \frac{d}{d x} [x^{2} - 7 x - 1]$

Step 1.3

Replace all occurrences of $u$ with $x^{2} - 7 x - 1$ .

$- 8 {(x^{2} - 7 x - 1)}^{- 9} \frac{d}{d x} [x^{2} - 7 x - 1]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} - 7 x - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 1]$ .

$- 8 {(x^{2} - 7 x - 1)}^{- 9} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 1])$

Step 2.2

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

$- 8 {(x^{2} - 7 x - 1)}^{- 9} (2 x + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 1])$

Step 2.3

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$- 8 {(x^{2} - 7 x - 1)}^{- 9} (2 x - 7 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 2.4

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

$- 8 {(x^{2} - 7 x - 1)}^{- 9} (2 x - 7 \cdot 1 + \frac{d}{d x} [- 1])$

Step 2.5

Multiply $- 7$ by $1$ .

$- 8 {(x^{2} - 7 x - 1)}^{- 9} (2 x - 7 + \frac{d}{d x} [- 1])$

Step 2.6

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

$- 8 {(x^{2} - 7 x - 1)}^{- 9} (2 x - 7 + 0)$

Step 2.7

Add $2 x - 7$ and $0$ .

$- 8 {(x^{2} - 7 x - 1)}^{- 9} (2 x - 7)$

Step 3

Simplify.

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Step 3.1

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

$- 8 \frac{1}{{(x^{2} - 7 x - 1)}^{9}} (2 x - 7)$

Step 3.2

Combine terms.

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Step 3.2.1

Combine $- 8$ and $\frac{1}{{(x^{2} - 7 x - 1)}^{9}}$ .

$\frac{- 8}{{(x^{2} - 7 x - 1)}^{9}} (2 x - 7)$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{8}{{(x^{2} - 7 x - 1)}^{9}} (2 x - 7)$

Step 3.3

Reorder the factors of $- \frac{8}{{(x^{2} - 7 x - 1)}^{9}} (2 x - 7)$ .

$- (2 x - 7) \frac{8}{{(x^{2} - 7 x - 1)}^{9}}$

Step 3.4

Apply the distributive property.

$(- (2 x) - - 7) \frac{8}{{(x^{2} - 7 x - 1)}^{9}}$

Step 3.5

Multiply $2$ by $- 1$ .

$(- 2 x - - 7) \frac{8}{{(x^{2} - 7 x - 1)}^{9}}$

Step 3.6

Multiply $- 1$ by $- 7$ .

$(- 2 x + 7) \frac{8}{{(x^{2} - 7 x - 1)}^{9}}$

Step 3.7

Multiply $- 2 x + 7$ by $\frac{8}{{(x^{2} - 7 x - 1)}^{9}}$ .

$\frac{(- 2 x + 7) \cdot 8}{{(x^{2} - 7 x - 1)}^{9}}$

Step 3.8

Move $8$ to the left of $- 2 x + 7$ .

$\frac{8 \cdot (- 2 x + 7)}{{(x^{2} - 7 x - 1)}^{9}}$

Step 3.9

Factor $- 1$ out of $- 2 x$ .

$\frac{8 (- (2 x) + 7)}{{(x^{2} - 7 x - 1)}^{9}}$

Step 3.10

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{8 (- (2 x) - 1 (- 7))}{{(x^{2} - 7 x - 1)}^{9}}$

Step 3.11

Factor $- 1$ out of $- (2 x) - 1 (- 7)$ .

$\frac{8 (- (2 x - 7))}{{(x^{2} - 7 x - 1)}^{9}}$

Step 3.12

Rewrite $- (2 x - 7)$ as $- 1 (2 x - 7)$ .

$\frac{8 (- 1 (2 x - 7))}{{(x^{2} - 7 x - 1)}^{9}}$

Step 3.13

Move the negative in front of the fraction.

$- \frac{8 (2 x - 7)}{{(x^{2} - 7 x - 1)}^{9}}$