Calculus Examples

$j (x) = \frac{4 x + 11}{x^{2} - 3}$

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 $f (x) = 4 x + 11$ and $g (x) = x^{2} - 3$ .

$\frac{(x^{2} - 3) \frac{d}{d x} [4 x + 11] - (4 x + 11) \frac{d}{d x} [x^{2} - 3]}{{(x^{2} - 3)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $4 x + 11$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [11]$ .

$\frac{(x^{2} - 3) (\frac{d}{d x} [4 x] + \frac{d}{d x} [11]) - (4 x + 11) \frac{d}{d x} [x^{2} - 3]}{{(x^{2} - 3)}^{2}}$

Step 2.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{(x^{2} - 3) (4 \frac{d}{d x} [x] + \frac{d}{d x} [11]) - (4 x + 11) \frac{d}{d x} [x^{2} - 3]}{{(x^{2} - 3)}^{2}}$

Step 2.3

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

$\frac{(x^{2} - 3) (4 \cdot 1 + \frac{d}{d x} [11]) - (4 x + 11) \frac{d}{d x} [x^{2} - 3]}{{(x^{2} - 3)}^{2}}$

Step 2.4

Multiply $4$ by $1$ .

$\frac{(x^{2} - 3) (4 + \frac{d}{d x} [11]) - (4 x + 11) \frac{d}{d x} [x^{2} - 3]}{{(x^{2} - 3)}^{2}}$

Step 2.5

Since $11$ is constant with respect to $x$ , the derivative of $11$ with respect to $x$ is $0$ .

$\frac{(x^{2} - 3) (4 + 0) - (4 x + 11) \frac{d}{d x} [x^{2} - 3]}{{(x^{2} - 3)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $4$ and $0$ .

$\frac{(x^{2} - 3) \cdot 4 - (4 x + 11) \frac{d}{d x} [x^{2} - 3]}{{(x^{2} - 3)}^{2}}$

Step 2.6.2

Move $4$ to the left of $x^{2} - 3$ .

$\frac{4 \cdot (x^{2} - 3) - (4 x + 11) \frac{d}{d x} [x^{2} - 3]}{{(x^{2} - 3)}^{2}}$

Step 2.7

By the Sum Rule, the derivative of $x^{2} - 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3]$ .

$\frac{4 (x^{2} - 3) - (4 x + 11) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3])}{{(x^{2} - 3)}^{2}}$

Step 2.8

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

$\frac{4 (x^{2} - 3) - (4 x + 11) (2 x + \frac{d}{d x} [- 3])}{{(x^{2} - 3)}^{2}}$

Step 2.9

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{4 (x^{2} - 3) - (4 x + 11) (2 x + 0)}{{(x^{2} - 3)}^{2}}$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $2 x$ and $0$ .

$\frac{4 (x^{2} - 3) - (4 x + 11) (2 x)}{{(x^{2} - 3)}^{2}}$

Step 2.10.2

Multiply $2$ by $- 1$ .

$\frac{4 (x^{2} - 3) - 2 (4 x + 11) x}{{(x^{2} - 3)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{4 x^{2} + 4 \cdot - 3 - 2 (4 x + 11) x}{{(x^{2} - 3)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{4 x^{2} + 4 \cdot - 3 + (- 2 (4 x) - 2 \cdot 11) x}{{(x^{2} - 3)}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{4 x^{2} + 4 \cdot - 3 - 2 (4 x) x - 2 \cdot 11 x}{{(x^{2} - 3)}^{2}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Multiply $4$ by $- 3$ .

$\frac{4 x^{2} - 12 - 2 (4 x) x - 2 \cdot 11 x}{{(x^{2} - 3)}^{2}}$

Step 3.4.1.2

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

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Step 3.4.1.2.1

Move $x$ .

$\frac{4 x^{2} - 12 - 2 (4 (x \cdot x)) - 2 \cdot 11 x}{{(x^{2} - 3)}^{2}}$

Step 3.4.1.2.2

Multiply $x$ by $x$ .

$\frac{4 x^{2} - 12 - 2 (4 x^{2}) - 2 \cdot 11 x}{{(x^{2} - 3)}^{2}}$

Step 3.4.1.3

Multiply $4$ by $- 2$ .

$\frac{4 x^{2} - 12 - 8 x^{2} - 2 \cdot 11 x}{{(x^{2} - 3)}^{2}}$

Step 3.4.1.4

Multiply $- 2$ by $11$ .

$\frac{4 x^{2} - 12 - 8 x^{2} - 22 x}{{(x^{2} - 3)}^{2}}$

Step 3.4.2

Subtract $8 x^{2}$ from $4 x^{2}$ .

$\frac{- 4 x^{2} - 12 - 22 x}{{(x^{2} - 3)}^{2}}$

Step 3.5

Reorder terms.

$\frac{- 4 x^{2} - 22 x - 12}{{(x^{2} - 3)}^{2}}$

Step 3.6

Factor $2$ out of $- 4 x^{2} - 22 x - 12$ .

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Step 3.6.1

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

$\frac{2 (- 2 x^{2}) - 22 x - 12}{{(x^{2} - 3)}^{2}}$

Step 3.6.2

Factor $2$ out of $- 22 x$ .

$\frac{2 (- 2 x^{2}) + 2 (- 11 x) - 12}{{(x^{2} - 3)}^{2}}$

Step 3.6.3

Factor $2$ out of $- 12$ .

$\frac{2 (- 2 x^{2}) + 2 (- 11 x) + 2 (- 6)}{{(x^{2} - 3)}^{2}}$

Step 3.6.4

Factor $2$ out of $2 (- 2 x^{2}) + 2 (- 11 x)$ .

$\frac{2 (- 2 x^{2} - 11 x) + 2 (- 6)}{{(x^{2} - 3)}^{2}}$

Step 3.6.5

Factor $2$ out of $2 (- 2 x^{2} - 11 x) + 2 (- 6)$ .

$\frac{2 (- 2 x^{2} - 11 x - 6)}{{(x^{2} - 3)}^{2}}$

Step 3.7

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

$\frac{2 (- (2 x^{2}) - 11 x - 6)}{{(x^{2} - 3)}^{2}}$

Step 3.8

Factor $- 1$ out of $- 11 x$ .

$\frac{2 (- (2 x^{2}) - (11 x) - 6)}{{(x^{2} - 3)}^{2}}$

Step 3.9

Factor $- 1$ out of $- (2 x^{2}) - (11 x)$ .

$\frac{2 (- (2 x^{2} + 11 x) - 6)}{{(x^{2} - 3)}^{2}}$

Step 3.10

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{2 (- (2 x^{2} + 11 x) - 1 (6))}{{(x^{2} - 3)}^{2}}$

Step 3.11

Factor $- 1$ out of $- (2 x^{2} + 11 x) - 1 (6)$ .

$\frac{2 (- (2 x^{2} + 11 x + 6))}{{(x^{2} - 3)}^{2}}$

Step 3.12

Rewrite $- (2 x^{2} + 11 x + 6)$ as $- 1 (2 x^{2} + 11 x + 6)$ .

$\frac{2 (- 1 (2 x^{2} + 11 x + 6))}{{(x^{2} - 3)}^{2}}$

Step 3.13

Move the negative in front of the fraction.

$- \frac{2 (2 x^{2} + 11 x + 6)}{{(x^{2} - 3)}^{2}}$