Calculus Examples

$\frac{- 16 x}{{(x^{2} - 4)}^{2}}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{16 x}{{(x^{2} - 4)}^{2}}]$

Step 1.2

Since $- 16$ is constant with respect to $x$ , the derivative of $- \frac{16 x}{{(x^{2} - 4)}^{2}}$ with respect to $x$ is $- 16 \frac{d}{d x} [\frac{x}{{(x^{2} - 4)}^{2}}]$ .

$- 16 \frac{d}{d x} [\frac{x}{{(x^{2} - 4)}^{2}}]$

Step 2

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

$- 16 \frac{{(x^{2} - 4)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} - 4)}^{2}]}{{({(x^{2} - 4)}^{2})}^{2}}$

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

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Step 3.1.1

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

$- 16 \frac{{(x^{2} - 4)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} - 4)}^{2}]}{{(x^{2} - 4)}^{2 \cdot 2}}$

Step 3.1.2

Multiply $2$ by $2$ .

$- 16 \frac{{(x^{2} - 4)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} - 4)}^{2}]}{{(x^{2} - 4)}^{4}}$

Step 3.2

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

$- 16 \frac{{(x^{2} - 4)}^{2} \cdot 1 - x \frac{d}{d x} [{(x^{2} - 4)}^{2}]}{{(x^{2} - 4)}^{4}}$

Step 3.3

Multiply ${(x^{2} - 4)}^{2}$ by $1$ .

$- 16 \frac{{(x^{2} - 4)}^{2} - x \frac{d}{d x} [{(x^{2} - 4)}^{2}]}{{(x^{2} - 4)}^{4}}$

Step 4

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 $g (x) = x^{2} - 4$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $x^{2} - 4$ .

$- 16 \frac{{(x^{2} - 4)}^{2} - x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} - 4])}{{(x^{2} - 4)}^{4}}$

Step 4.2

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

$- 16 \frac{{(x^{2} - 4)}^{2} - x (2 u \frac{d}{d x} [x^{2} - 4])}{{(x^{2} - 4)}^{4}}$

Step 4.3

Replace all occurrences of $u$ with $x^{2} - 4$ .

$- 16 \frac{{(x^{2} - 4)}^{2} - x (2 (x^{2} - 4) \frac{d}{d x} [x^{2} - 4])}{{(x^{2} - 4)}^{4}}$

Step 5

Simplify with factoring out.

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Step 5.1

Multiply $2$ by $- 1$ .

$- 16 \frac{{(x^{2} - 4)}^{2} - 2 x ((x^{2} - 4) \frac{d}{d x} [x^{2} - 4])}{{(x^{2} - 4)}^{4}}$

Step 5.2

Factor $x^{2} - 4$ out of ${(x^{2} - 4)}^{2} - 2 x ((x^{2} - 4) \frac{d}{d x} [x^{2} - 4])$ .

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Step 5.2.1

Factor $x^{2} - 4$ out of ${(x^{2} - 4)}^{2}$ .

$- 16 \frac{(x^{2} - 4) (x^{2} - 4) - 2 x ((x^{2} - 4) \frac{d}{d x} [x^{2} - 4])}{{(x^{2} - 4)}^{4}}$

Step 5.2.2

Factor $x^{2} - 4$ out of $- 2 x ((x^{2} - 4) \frac{d}{d x} [x^{2} - 4])$ .

$- 16 \frac{(x^{2} - 4) (x^{2} - 4) + (x^{2} - 4) (- 2 x (\frac{d}{d x} [x^{2} - 4]))}{{(x^{2} - 4)}^{4}}$

Step 5.2.3

Factor $x^{2} - 4$ out of $(x^{2} - 4) (x^{2} - 4) + (x^{2} - 4) (- 2 x (\frac{d}{d x} [x^{2} - 4]))$ .

$- 16 \frac{(x^{2} - 4) (x^{2} - 4 - 2 x (\frac{d}{d x} [x^{2} - 4]))}{{(x^{2} - 4)}^{4}}$

Step 6

Cancel the common factors.

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Step 6.1

Factor $x^{2} - 4$ out of ${(x^{2} - 4)}^{4}$ .

$- 16 \frac{(x^{2} - 4) (x^{2} - 4 - 2 x (\frac{d}{d x} [x^{2} - 4]))}{(x^{2} - 4) {(x^{2} - 4)}^{3}}$

Step 6.2

Cancel the common factor.

$- 16 \frac{(x^{2} - 4) (x^{2} - 4 - 2 x (\frac{d}{d x} [x^{2} - 4]))}{(x^{2} - 4) {(x^{2} - 4)}^{3}}$

Step 6.3

Rewrite the expression.

$- 16 \frac{x^{2} - 4 - 2 x (\frac{d}{d x} [x^{2} - 4])}{{(x^{2} - 4)}^{3}}$

Step 7

By the Sum Rule, the derivative of $x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$ .

$- 16 \frac{x^{2} - 4 - 2 x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4])}{{(x^{2} - 4)}^{3}}$

Step 8

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

$- 16 \frac{x^{2} - 4 - 2 x (2 x + \frac{d}{d x} [- 4])}{{(x^{2} - 4)}^{3}}$

Step 9

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$- 16 \frac{x^{2} - 4 - 2 x (2 x + 0)}{{(x^{2} - 4)}^{3}}$

Step 10

Simplify the expression.

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Step 10.1

Add $2 x$ and $0$ .

$- 16 \frac{x^{2} - 4 - 2 x (2 x)}{{(x^{2} - 4)}^{3}}$

Step 10.2

Multiply $2$ by $- 2$ .

$- 16 \frac{x^{2} - 4 - 4 x \cdot x}{{(x^{2} - 4)}^{3}}$

Step 11

Raise $x$ to the power of $1$ .

$- 16 \frac{x^{2} - 4 - 4 (x^{1} x)}{{(x^{2} - 4)}^{3}}$

Step 12

Raise $x$ to the power of $1$ .

$- 16 \frac{x^{2} - 4 - 4 (x^{1} x^{1})}{{(x^{2} - 4)}^{3}}$

Step 13

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 16 \frac{x^{2} - 4 - 4 x^{1 + 1}}{{(x^{2} - 4)}^{3}}$

Step 14

Add $1$ and $1$ .

$- 16 \frac{x^{2} - 4 - 4 x^{2}}{{(x^{2} - 4)}^{3}}$

Step 15

Subtract $4 x^{2}$ from $x^{2}$ .

$- 16 \frac{- 3 x^{2} - 4}{{(x^{2} - 4)}^{3}}$

Step 16

Combine $- 16$ and $\frac{- 3 x^{2} - 4}{{(x^{2} - 4)}^{3}}$ .

$\frac{- 16 (- 3 x^{2} - 4)}{{(x^{2} - 4)}^{3}}$

Step 17

Move the negative in front of the fraction.

$- \frac{16 (- 3 x^{2} - 4)}{{(x^{2} - 4)}^{3}}$

Step 18

Simplify.

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Step 18.1

Apply the distributive property.

$- \frac{16 (- 3 x^{2}) + 16 \cdot - 4}{{(x^{2} - 4)}^{3}}$

Step 18.2

Simplify each term.

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Step 18.2.1

Multiply $- 3$ by $16$ .

$- \frac{- 48 x^{2} + 16 \cdot - 4}{{(x^{2} - 4)}^{3}}$

Step 18.2.2

Multiply $16$ by $- 4$ .

$- \frac{- 48 x^{2} - 64}{{(x^{2} - 4)}^{3}}$