Calculus Examples

$f (x) = \frac{5 x - 7}{x^{2} - 2}$

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 $f (x) = 5 x - 7$ and $g (x) = x^{2} - 2$ .

$\frac{(x^{2} - 2) \frac{d}{d x} [5 x - 7] - (5 x - 7) \frac{d}{d x} [x^{2} - 2]}{{(x^{2} - 2)}^{2}}$

Step 2

By the Sum Rule, the derivative of $5 x - 7$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 7]$ .

$\frac{(x^{2} - 2) (\frac{d}{d x} [5 x] + \frac{d}{d x} [- 7]) - (5 x - 7) \frac{d}{d x} [x^{2} - 2]}{{(x^{2} - 2)}^{2}}$

Step 3

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\frac{(x^{2} - 2) (5 \frac{d}{d x} [x] + \frac{d}{d x} [- 7]) - (5 x - 7) \frac{d}{d x} [x^{2} - 2]}{{(x^{2} - 2)}^{2}}$

Step 3.2

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

$\frac{(x^{2} - 2) (5 \cdot 1 + \frac{d}{d x} [- 7]) - (5 x - 7) \frac{d}{d x} [x^{2} - 2]}{{(x^{2} - 2)}^{2}}$

Step 3.3

Multiply $5$ by $1$ .

$\frac{(x^{2} - 2) (5 + \frac{d}{d x} [- 7]) - (5 x - 7) \frac{d}{d x} [x^{2} - 2]}{{(x^{2} - 2)}^{2}}$

Step 4

Differentiate.

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Step 4.1

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

$\frac{(x^{2} - 2) (5 + 0) - (5 x - 7) \frac{d}{d x} [x^{2} - 2]}{{(x^{2} - 2)}^{2}}$

Step 4.2

Add $5$ and $0$ .

$\frac{(x^{2} - 2) \cdot 5 - (5 x - 7) \frac{d}{d x} [x^{2} - 2]}{{(x^{2} - 2)}^{2}}$

Step 4.3

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

$\frac{(x^{2} - 2) \cdot 5 - (5 x - 7) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2])}{{(x^{2} - 2)}^{2}}$

Step 4.4

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

$\frac{(x^{2} - 2) \cdot 5 - (5 x - 7) (2 x + \frac{d}{d x} [- 2])}{{(x^{2} - 2)}^{2}}$

Step 4.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$\frac{(x^{2} - 2) \cdot 5 - (5 x - 7) (2 x + 0)}{{(x^{2} - 2)}^{2}}$

Step 4.6

Add $2 x$ and $0$ .

$\frac{(x^{2} - 2) \cdot 5 - (5 x - 7) (2 x)}{{(x^{2} - 2)}^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{x^{2} \cdot 5 - 2 \cdot 5 - (5 x - 7) (2 x)}{{(x^{2} - 2)}^{2}}$

Step 5.2

Apply the distributive property.

$\frac{x^{2} \cdot 5 - 2 \cdot 5 + (- (5 x) - - 7) (2 x)}{{(x^{2} - 2)}^{2}}$

Step 5.3

Apply the distributive property.

$\frac{x^{2} \cdot 5 - 2 \cdot 5 - (5 x) (2 x) - - 7 (2 x)}{{(x^{2} - 2)}^{2}}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Move $5$ to the left of $x^{2}$ .

$\frac{5 \cdot x^{2} - 2 \cdot 5 - (5 x) (2 x) - - 7 (2 x)}{{(x^{2} - 2)}^{2}}$

Step 5.4.1.2

Multiply $- 2$ by $5$ .

$\frac{5 x^{2} - 10 - (5 x) (2 x) - - 7 (2 x)}{{(x^{2} - 2)}^{2}}$

Step 5.4.1.3

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

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Step 5.4.1.3.1

Move $x$ .

$\frac{5 x^{2} - 10 - (5 (x \cdot x)) \cdot 2 - - 7 (2 x)}{{(x^{2} - 2)}^{2}}$

Step 5.4.1.3.2

Multiply $x$ by $x$ .

$\frac{5 x^{2} - 10 - (5 x^{2}) \cdot 2 - - 7 (2 x)}{{(x^{2} - 2)}^{2}}$

Step 5.4.1.4

Multiply $5$ by $- 1$ .

$\frac{5 x^{2} - 10 - 5 x^{2} \cdot 2 - - 7 (2 x)}{{(x^{2} - 2)}^{2}}$

Step 5.4.1.5

Multiply $2$ by $- 5$ .

$\frac{5 x^{2} - 10 - 10 x^{2} - - 7 (2 x)}{{(x^{2} - 2)}^{2}}$

Step 5.4.1.6

Multiply $- 1$ by $- 7$ .

$\frac{5 x^{2} - 10 - 10 x^{2} + 7 (2 x)}{{(x^{2} - 2)}^{2}}$

Step 5.4.1.7

Multiply $2$ by $7$ .

$\frac{5 x^{2} - 10 - 10 x^{2} + 14 x}{{(x^{2} - 2)}^{2}}$

Step 5.4.2

Subtract $10 x^{2}$ from $5 x^{2}$ .

$\frac{- 5 x^{2} - 10 + 14 x}{{(x^{2} - 2)}^{2}}$

Step 5.5

Reorder terms.

$\frac{- 5 x^{2} + 14 x - 10}{{(x^{2} - 2)}^{2}}$

Step 5.6

Factor $- 1$ out of $- 5 x^{2}$ .

$\frac{- (5 x^{2}) + 14 x - 10}{{(x^{2} - 2)}^{2}}$

Step 5.7

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

$\frac{- (5 x^{2}) - (- 14 x) - 10}{{(x^{2} - 2)}^{2}}$

Step 5.8

Factor $- 1$ out of $- (5 x^{2}) - (- 14 x)$ .

$\frac{- (5 x^{2} - 14 x) - 10}{{(x^{2} - 2)}^{2}}$

Step 5.9

Rewrite $- 10$ as $- 1 (10)$ .

$\frac{- (5 x^{2} - 14 x) - 1 (10)}{{(x^{2} - 2)}^{2}}$

Step 5.10

Factor $- 1$ out of $- (5 x^{2} - 14 x) - 1 (10)$ .

$\frac{- (5 x^{2} - 14 x + 10)}{{(x^{2} - 2)}^{2}}$

Step 5.11

Rewrite $- (5 x^{2} - 14 x + 10)$ as $- 1 (5 x^{2} - 14 x + 10)$ .

$\frac{- 1 (5 x^{2} - 14 x + 10)}{{(x^{2} - 2)}^{2}}$

Step 5.12

Move the negative in front of the fraction.

$- \frac{5 x^{2} - 14 x + 10}{{(x^{2} - 2)}^{2}}$