Calculus Examples

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

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 $f (x) = x + 4$ and $g (x) = (x - 1) (x + 3)$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

$\frac{(x - 1) (x + 3) (1 + 0) - (x + 4) \frac{d}{d x} [(x - 1) (x + 3)]}{{((x - 1) (x + 3))}^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $1$ and $0$ .

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

Step 2.4.2

Multiply $x - 1$ by $1$ .

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

Step 3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x - 1$ and $g (x) = x + 3$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

$\frac{(x - 1) (x + 3) - (x + 4) ((x - 1) (1 + 0) + (x + 3) \frac{d}{d x} [x - 1])}{{((x - 1) (x + 3))}^{2}}$

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $1$ and $0$ .

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

Step 4.4.2

Multiply $x - 1$ by $1$ .

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

Step 4.5

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 4.6

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

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

Step 4.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{(x - 1) (x + 3) - (x + 4) (x - 1 + (x + 3) (1 + 0))}{{((x - 1) (x + 3))}^{2}}$

Step 4.8

Simplify by adding terms.

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Step 4.8.1

Add $1$ and $0$ .

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

Step 4.8.2

Multiply $x + 3$ by $1$ .

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

Step 4.8.3

Add $x$ and $x$ .

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

Step 4.8.4

Add $- 1$ and $3$ .

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

Step 5

Simplify.

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Step 5.1

Apply the product rule to $(x - 1) (x + 3)$ .

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

Step 5.2

Apply the distributive property.

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

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

Expand $(x - 1) (x + 3)$ using the FOIL Method.

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Step 5.3.1.1.1

Apply the distributive property.

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

Step 5.3.1.1.2

Apply the distributive property.

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

Step 5.3.1.1.3

Apply the distributive property.

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

Step 5.3.1.2

Simplify and combine like terms.

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Step 5.3.1.2.1

Simplify each term.

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Step 5.3.1.2.1.1

Multiply $x$ by $x$ .

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

Step 5.3.1.2.1.2

Move $3$ to the left of $x$ .

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

Step 5.3.1.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 5.3.1.2.1.4

Multiply $- 1$ by $3$ .

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

Step 5.3.1.2.2

Subtract $x$ from $3 x$ .

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

Step 5.3.1.3

Multiply $- 1$ by $4$ .

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

Step 5.3.1.4

Expand $(- x - 4) (2 x + 2)$ using the FOIL Method.

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Step 5.3.1.4.1

Apply the distributive property.

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

Step 5.3.1.4.2

Apply the distributive property.

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

Step 5.3.1.4.3

Apply the distributive property.

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

Step 5.3.1.5

Simplify and combine like terms.

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Step 5.3.1.5.1

Simplify each term.

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Step 5.3.1.5.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.3.1.5.1.2

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

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Step 5.3.1.5.1.2.1

Move $x$ .

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

Step 5.3.1.5.1.2.2

Multiply $x$ by $x$ .

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

Step 5.3.1.5.1.3

Multiply $- 1$ by $2$ .

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

Step 5.3.1.5.1.4

Multiply $2$ by $- 1$ .

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

Step 5.3.1.5.1.5

Multiply $2$ by $- 4$ .

$\frac{x^{2} + 2 x - 3 - 2 x^{2} - 2 x - 8 x - 4 \cdot 2}{{(x - 1)}^{2} {(x + 3)}^{2}}$

Step 5.3.1.5.1.6

Multiply $- 4$ by $2$ .

$\frac{x^{2} + 2 x - 3 - 2 x^{2} - 2 x - 8 x - 8}{{(x - 1)}^{2} {(x + 3)}^{2}}$

Step 5.3.1.5.2

Subtract $8 x$ from $- 2 x$ .

$\frac{x^{2} + 2 x - 3 - 2 x^{2} - 10 x - 8}{{(x - 1)}^{2} {(x + 3)}^{2}}$

Step 5.3.2

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

$\frac{- x^{2} + 2 x - 3 - 10 x - 8}{{(x - 1)}^{2} {(x + 3)}^{2}}$

Step 5.3.3

Subtract $10 x$ from $2 x$ .

$\frac{- x^{2} - 8 x - 3 - 8}{{(x - 1)}^{2} {(x + 3)}^{2}}$

Step 5.3.4

Subtract $8$ from $- 3$ .

$\frac{- x^{2} - 8 x - 11}{{(x - 1)}^{2} {(x + 3)}^{2}}$

Step 5.4

Reorder terms.

$\frac{- x^{2} - 8 x - 11}{{(x + 3)}^{2} {(x - 1)}^{2}}$

Step 5.5

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

$\frac{- (x^{2}) - 8 x - 11}{{(x + 3)}^{2} {(x - 1)}^{2}}$

Step 5.6

Factor $- 1$ out of $- 8 x$ .

$\frac{- (x^{2}) - (8 x) - 11}{{(x + 3)}^{2} {(x - 1)}^{2}}$

Step 5.7

Factor $- 1$ out of $- (x^{2}) - (8 x)$ .

$\frac{- (x^{2} + 8 x) - 11}{{(x + 3)}^{2} {(x - 1)}^{2}}$

Step 5.8

Rewrite $- 11$ as $- 1 (11)$ .

$\frac{- (x^{2} + 8 x) - 1 (11)}{{(x + 3)}^{2} {(x - 1)}^{2}}$

Step 5.9

Factor $- 1$ out of $- (x^{2} + 8 x) - 1 (11)$ .

$\frac{- (x^{2} + 8 x + 11)}{{(x + 3)}^{2} {(x - 1)}^{2}}$

Step 5.10

Rewrite $- (x^{2} + 8 x + 11)$ as $- 1 (x^{2} + 8 x + 11)$ .

$\frac{- 1 (x^{2} + 8 x + 11)}{{(x + 3)}^{2} {(x - 1)}^{2}}$

Step 5.11

Move the negative in front of the fraction.

$- \frac{x^{2} + 8 x + 11}{{(x + 3)}^{2} {(x - 1)}^{2}}$