Calculus Examples

$\frac{8 x - 16}{x^{2}}$

Step 1

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 $f (x) = 8 x - 16$ and $g (x) = x^{2}$ .

$\frac{x^{2} \frac{d}{d x} [8 x - 16] - (8 x - 16) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 2

By the Sum Rule, the derivative of $8 x - 16$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [- 16]$ .

$\frac{x^{2} (\frac{d}{d x} [8 x] + \frac{d}{d x} [- 16]) - (8 x - 16) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 3

Evaluate $\frac{d}{d x} [8 x]$ .

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Step 3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$\frac{x^{2} (8 \frac{d}{d x} [x] + \frac{d}{d x} [- 16]) - (8 x - 16) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 3.2

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

$\frac{x^{2} (8 \cdot 1 + \frac{d}{d x} [- 16]) - (8 x - 16) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 3.3

Multiply $8$ by $1$ .

$\frac{x^{2} (8 + \frac{d}{d x} [- 16]) - (8 x - 16) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 4

Differentiate.

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Step 4.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$\frac{x^{2} (8 + 0) - (8 x - 16) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 4.2

Add $8$ and $0$ .

$\frac{x^{2} \cdot 8 - (8 x - 16) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 4.3

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

$\frac{x^{2} \cdot 8 - (8 x - 16) (2 x)}{{(x^{2})}^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{x^{2} \cdot 8 + (- (8 x) - - 16) (2 x)}{{(x^{2})}^{2}}$

Step 5.2

Apply the distributive property.

$\frac{x^{2} \cdot 8 - (8 x) (2 x) - - 16 (2 x)}{{(x^{2})}^{2}}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Move $8$ to the left of $x^{2}$ .

$\frac{8 \cdot x^{2} - (8 x) (2 x) - - 16 (2 x)}{{(x^{2})}^{2}}$

Step 5.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.1.2.1

Move $x$ .

$\frac{8 x^{2} - (8 (x \cdot x)) \cdot 2 - - 16 (2 x)}{{(x^{2})}^{2}}$

Step 5.3.1.2.2

Multiply $x$ by $x$ .

$\frac{8 x^{2} - (8 x^{2}) \cdot 2 - - 16 (2 x)}{{(x^{2})}^{2}}$

Step 5.3.1.3

Multiply $8$ by $- 1$ .

$\frac{8 x^{2} - 8 x^{2} \cdot 2 - - 16 (2 x)}{{(x^{2})}^{2}}$

Step 5.3.1.4

Multiply $2$ by $- 8$ .

$\frac{8 x^{2} - 16 x^{2} - - 16 (2 x)}{{(x^{2})}^{2}}$

Step 5.3.1.5

Multiply $- 1$ by $- 16$ .

$\frac{8 x^{2} - 16 x^{2} + 16 (2 x)}{{(x^{2})}^{2}}$

Step 5.3.1.6

Multiply $2$ by $16$ .

$\frac{8 x^{2} - 16 x^{2} + 32 x}{{(x^{2})}^{2}}$

Step 5.3.2

Subtract $16 x^{2}$ from $8 x^{2}$ .

$\frac{- 8 x^{2} + 32 x}{{(x^{2})}^{2}}$

Step 5.4

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

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Step 5.4.1

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

$\frac{- 8 x^{2} + 32 x}{x^{2 \cdot 2}}$

Step 5.4.2

Multiply $2$ by $2$ .

$\frac{- 8 x^{2} + 32 x}{x^{4}}$

Step 5.5

Factor $8 x$ out of $- 8 x^{2} + 32 x$ .

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Step 5.5.1

Factor $8 x$ out of $- 8 x^{2}$ .

$\frac{8 x (- x) + 32 x}{x^{4}}$

Step 5.5.2

Factor $8 x$ out of $32 x$ .

$\frac{8 x (- x) + 8 x (4)}{x^{4}}$

Step 5.5.3

Factor $8 x$ out of $8 x (- x) + 8 x (4)$ .

$\frac{8 x (- x + 4)}{x^{4}}$

Step 5.6

Cancel the common factor of $x$ and $x^{4}$ .

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Step 5.6.1

Factor $x$ out of $8 x (- x + 4)$ .

$\frac{x (8 (- x + 4))}{x^{4}}$

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

Factor $x$ out of $x^{4}$ .

$\frac{x (8 (- x + 4))}{x \cdot x^{3}}$

Step 5.6.2.2

Cancel the common factor.

$\frac{x (8 (- x + 4))}{x \cdot x^{3}}$

Step 5.6.2.3

Rewrite the expression.

$\frac{8 (- x + 4)}{x^{3}}$

Step 5.7

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

$\frac{8 (- (x) + 4)}{x^{3}}$

Step 5.8

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{8 (- (x) - 1 (- 4))}{x^{3}}$

Step 5.9

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

$\frac{8 (- (x - 4))}{x^{3}}$

Step 5.10

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

$\frac{8 (- 1 (x - 4))}{x^{3}}$

Step 5.11

Move the negative in front of the fraction.

$- \frac{8 (x - 4)}{x^{3}}$