Calculus Examples

$f (x) = \frac{x - 7}{x^{2}}$

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 $f (x) = x - 7$ and $g (x) = x^{2}$ .

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

Step 2

Differentiate.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Multiply $2$ by $2$ .

$\frac{x^{2} \frac{d}{d x} [x - 7] - (x - 7) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2

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

$\frac{x^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [- 7]) - (x - 7) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.3

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

$\frac{x^{2} (1 + \frac{d}{d x} [- 7]) - (x - 7) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.4

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$\frac{x^{2} (1 + 0) - (x - 7) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.5

Simplify the expression.

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Step 2.5.1

Add $1$ and $0$ .

$\frac{x^{2} \cdot 1 - (x - 7) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.5.2

Multiply $x^{2}$ by $1$ .

$\frac{x^{2} - (x - 7) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.6

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

$\frac{x^{2} - (x - 7) (2 x)}{x^{4}}$

Step 2.7

Simplify with factoring out.

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Step 2.7.1

Multiply $2$ by $- 1$ .

$\frac{x^{2} - 2 (x - 7) x}{x^{4}}$

Step 2.7.2

Factor $x$ out of $x^{2} - 2 (x - 7) x$ .

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Step 2.7.2.1

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

$\frac{x \cdot x - 2 (x - 7) x}{x^{4}}$

Step 2.7.2.2

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

$\frac{x \cdot x + x (- 2 (x - 7))}{x^{4}}$

Step 2.7.2.3

Factor $x$ out of $x \cdot x + x (- 2 (x - 7))$ .

$\frac{x (x - 2 (x - 7))}{x^{4}}$

Step 3

Cancel the common factors.

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Step 3.1

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

$\frac{x (x - 2 (x - 7))}{x \cdot x^{3}}$

Step 3.2

Cancel the common factor.

$\frac{x (x - 2 (x - 7))}{x \cdot x^{3}}$

Step 3.3

Rewrite the expression.

$\frac{x - 2 (x - 7)}{x^{3}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{x - 2 x - 2 \cdot - 7}{x^{3}}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Multiply $- 2$ by $- 7$ .

$\frac{x - 2 x + 14}{x^{3}}$

Step 4.2.2

Subtract $2 x$ from $x$ .

$\frac{- x + 14}{x^{3}}$

Step 4.3

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

$\frac{- (x) + 14}{x^{3}}$

Step 4.4

Rewrite $14$ as $- 1 (- 14)$ .

$\frac{- (x) - 1 (- 14)}{x^{3}}$

Step 4.5

Factor $- 1$ out of $- (x) - 1 (- 14)$ .

$\frac{- (x - 14)}{x^{3}}$

Step 4.6

Rewrite $- (x - 14)$ as $- 1 (x - 14)$ .

$\frac{- 1 (x - 14)}{x^{3}}$

Step 4.7

Move the negative in front of the fraction.

$- \frac{x - 14}{x^{3}}$