Calculus Examples

$y = \frac{1}{{(x^{2} - 2 x - 5)}^{4}}$

Step 1

Apply basic rules of exponents.

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Step 1.1

Rewrite $\frac{1}{{(x^{2} - 2 x - 5)}^{4}}$ as ${({(x^{2} - 2 x - 5)}^{4})}^{- 1}$ .

$\frac{d}{d x} [{({(x^{2} - 2 x - 5)}^{4})}^{- 1}]$

Step 1.2

Multiply the exponents in ${({(x^{2} - 2 x - 5)}^{4})}^{- 1}$ .

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Step 1.2.1

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

$\frac{d}{d x} [{(x^{2} - 2 x - 5)}^{4 \cdot - 1}]$

Step 1.2.2

Multiply $4$ by $- 1$ .

$\frac{d}{d x} [{(x^{2} - 2 x - 5)}^{- 4}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 4}$ and $g (x) = x^{2} - 2 x - 5$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{2} - 2 x - 5$ .

$\frac{d}{d u} [u^{- 4}] \frac{d}{d x} [x^{2} - 2 x - 5]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 4$ .

$- 4 u^{- 5} \frac{d}{d x} [x^{2} - 2 x - 5]$

Step 2.3

Replace all occurrences of $u$ with $x^{2} - 2 x - 5$ .

$- 4 {(x^{2} - 2 x - 5)}^{- 5} \frac{d}{d x} [x^{2} - 2 x - 5]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x^{2} - 2 x - 5$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 5]$ .

$- 4 {(x^{2} - 2 x - 5)}^{- 5} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 5])$

Step 3.2

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

$- 4 {(x^{2} - 2 x - 5)}^{- 5} (2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 5])$

Step 3.3

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

$- 4 {(x^{2} - 2 x - 5)}^{- 5} (2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 5])$

Step 3.4

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

$- 4 {(x^{2} - 2 x - 5)}^{- 5} (2 x - 2 \cdot 1 + \frac{d}{d x} [- 5])$

Step 3.5

Multiply $- 2$ by $1$ .

$- 4 {(x^{2} - 2 x - 5)}^{- 5} (2 x - 2 + \frac{d}{d x} [- 5])$

Step 3.6

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$- 4 {(x^{2} - 2 x - 5)}^{- 5} (2 x - 2 + 0)$

Step 3.7

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

$- 4 {(x^{2} - 2 x - 5)}^{- 5} (2 x - 2)$

Step 4

Simplify.

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Step 4.1

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

$- 4 \frac{1}{{(x^{2} - 2 x - 5)}^{5}} (2 x - 2)$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $- 4$ and $\frac{1}{{(x^{2} - 2 x - 5)}^{5}}$ .

$\frac{- 4}{{(x^{2} - 2 x - 5)}^{5}} (2 x - 2)$

Step 4.2.2

Move the negative in front of the fraction.

$- \frac{4}{{(x^{2} - 2 x - 5)}^{5}} (2 x - 2)$

Step 4.3

Reorder the factors of $- \frac{4}{{(x^{2} - 2 x - 5)}^{5}} (2 x - 2)$ .

$- (2 x - 2) \frac{4}{{(x^{2} - 2 x - 5)}^{5}}$

Step 4.4

Apply the distributive property.

$(- (2 x) - - 2) \frac{4}{{(x^{2} - 2 x - 5)}^{5}}$

Step 4.5

Multiply $2$ by $- 1$ .

$(- 2 x - - 2) \frac{4}{{(x^{2} - 2 x - 5)}^{5}}$

Step 4.6

Multiply $- 1$ by $- 2$ .

$(- 2 x + 2) \frac{4}{{(x^{2} - 2 x - 5)}^{5}}$

Step 4.7

Multiply $- 2 x + 2$ by $\frac{4}{{(x^{2} - 2 x - 5)}^{5}}$ .

$\frac{(- 2 x + 2) \cdot 4}{{(x^{2} - 2 x - 5)}^{5}}$

Step 4.8

Simplify the numerator.

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Step 4.8.1

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

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Step 4.8.1.1

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

$\frac{(2 (- x) + 2) \cdot 4}{{(x^{2} - 2 x - 5)}^{5}}$

Step 4.8.1.2

Factor $2$ out of $2$ .

$\frac{(2 (- x) + 2 (1)) \cdot 4}{{(x^{2} - 2 x - 5)}^{5}}$

Step 4.8.1.3

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

$\frac{2 (- x + 1) \cdot 4}{{(x^{2} - 2 x - 5)}^{5}}$

Step 4.8.2

Multiply $4$ by $2$ .

$\frac{8 (- x + 1)}{{(x^{2} - 2 x - 5)}^{5}}$

Step 4.9

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

$\frac{8 (- (x) + 1)}{{(x^{2} - 2 x - 5)}^{5}}$

Step 4.10

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

$\frac{8 (- (x) - 1 (- 1))}{{(x^{2} - 2 x - 5)}^{5}}$

Step 4.11

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

$\frac{8 (- (x - 1))}{{(x^{2} - 2 x - 5)}^{5}}$

Step 4.12

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

$\frac{8 (- 1 (x - 1))}{{(x^{2} - 2 x - 5)}^{5}}$

Step 4.13

Move the negative in front of the fraction.

$- \frac{8 (x - 1)}{{(x^{2} - 2 x - 5)}^{5}}$