Calculus Examples

$f (x) = \frac{x^{2} - 7 x - 8}{x + 1}$

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 $f (x) = x^{2} - 7 x - 8$ and $g (x) = x + 1$ .

$\frac{(x + 1) \frac{d}{d x} [x^{2} - 7 x - 8] - (x^{2} - 7 x - 8) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} - 7 x - 8$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 8]$ .

$\frac{(x + 1) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 8]) - (x^{2} - 7 x - 8) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 2.2

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

$\frac{(x + 1) (2 x + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 8]) - (x^{2} - 7 x - 8) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 2.3

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$\frac{(x + 1) (2 x - 7 \frac{d}{d x} [x] + \frac{d}{d x} [- 8]) - (x^{2} - 7 x - 8) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 2.4

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

$\frac{(x + 1) (2 x - 7 \cdot 1 + \frac{d}{d x} [- 8]) - (x^{2} - 7 x - 8) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 2.5

Multiply $- 7$ by $1$ .

$\frac{(x + 1) (2 x - 7 + \frac{d}{d x} [- 8]) - (x^{2} - 7 x - 8) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 2.6

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$\frac{(x + 1) (2 x - 7 + 0) - (x^{2} - 7 x - 8) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 2.7

Add $2 x - 7$ and $0$ .

$\frac{(x + 1) (2 x - 7) - (x^{2} - 7 x - 8) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 2.8

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

$\frac{(x + 1) (2 x - 7) - (x^{2} - 7 x - 8) (\frac{d}{d x} [x] + \frac{d}{d x} [1])}{{(x + 1)}^{2}}$

Step 2.9

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

$\frac{(x + 1) (2 x - 7) - (x^{2} - 7 x - 8) (1 + \frac{d}{d x} [1])}{{(x + 1)}^{2}}$

Step 2.10

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

$\frac{(x + 1) (2 x - 7) - (x^{2} - 7 x - 8) (1 + 0)}{{(x + 1)}^{2}}$

Step 2.11

Simplify the expression.

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Step 2.11.1

Add $1$ and $0$ .

$\frac{(x + 1) (2 x - 7) - (x^{2} - 7 x - 8) \cdot 1}{{(x + 1)}^{2}}$

Step 2.11.2

Multiply $- 1$ by $1$ .

$\frac{(x + 1) (2 x - 7) - (x^{2} - 7 x - 8)}{{(x + 1)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(x + 1) (2 x - 7) - x^{2} - (- 7 x) - - 8}{{(x + 1)}^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Expand $(x + 1) (2 x - 7)$ using the FOIL Method.

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Step 3.2.1.1.1

Apply the distributive property.

$\frac{x (2 x - 7) + 1 (2 x - 7) - x^{2} - (- 7 x) - - 8}{{(x + 1)}^{2}}$

Step 3.2.1.1.2

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 7 + 1 (2 x - 7) - x^{2} - (- 7 x) - - 8}{{(x + 1)}^{2}}$

Step 3.2.1.1.3

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 7 + 1 (2 x) + 1 \cdot - 7 - x^{2} - (- 7 x) - - 8}{{(x + 1)}^{2}}$

Step 3.2.1.2

Simplify and combine like terms.

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Step 3.2.1.2.1

Simplify each term.

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Step 3.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x \cdot x + x \cdot - 7 + 1 (2 x) + 1 \cdot - 7 - x^{2} - (- 7 x) - - 8}{{(x + 1)}^{2}}$

Step 3.2.1.2.1.2

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

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Step 3.2.1.2.1.2.1

Move $x$ .

$\frac{2 (x \cdot x) + x \cdot - 7 + 1 (2 x) + 1 \cdot - 7 - x^{2} - (- 7 x) - - 8}{{(x + 1)}^{2}}$

Step 3.2.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + x \cdot - 7 + 1 (2 x) + 1 \cdot - 7 - x^{2} - (- 7 x) - - 8}{{(x + 1)}^{2}}$

Step 3.2.1.2.1.3

Move $- 7$ to the left of $x$ .

$\frac{2 x^{2} - 7 \cdot x + 1 (2 x) + 1 \cdot - 7 - x^{2} - (- 7 x) - - 8}{{(x + 1)}^{2}}$

Step 3.2.1.2.1.4

Multiply $2 x$ by $1$ .

$\frac{2 x^{2} - 7 x + 2 x + 1 \cdot - 7 - x^{2} - (- 7 x) - - 8}{{(x + 1)}^{2}}$

Step 3.2.1.2.1.5

Multiply $- 7$ by $1$ .

$\frac{2 x^{2} - 7 x + 2 x - 7 - x^{2} - (- 7 x) - - 8}{{(x + 1)}^{2}}$

Step 3.2.1.2.2

Add $- 7 x$ and $2 x$ .

$\frac{2 x^{2} - 5 x - 7 - x^{2} - (- 7 x) - - 8}{{(x + 1)}^{2}}$

Step 3.2.1.3

Multiply $- 7$ by $- 1$ .

$\frac{2 x^{2} - 5 x - 7 - x^{2} + 7 x - - 8}{{(x + 1)}^{2}}$

Step 3.2.1.4

Multiply $- 1$ by $- 8$ .

$\frac{2 x^{2} - 5 x - 7 - x^{2} + 7 x + 8}{{(x + 1)}^{2}}$

Step 3.2.2

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

$\frac{x^{2} - 5 x - 7 + 7 x + 8}{{(x + 1)}^{2}}$

Step 3.2.3

Add $- 5 x$ and $7 x$ .

$\frac{x^{2} + 2 x - 7 + 8}{{(x + 1)}^{2}}$

Step 3.2.4

Add $- 7$ and $8$ .

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

Step 3.3

Factor using the perfect square rule.

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Step 3.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 3.3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 3.3.3

Rewrite the polynomial.

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

Step 3.3.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 3.4

Cancel the common factor of ${(x + 1)}^{2}$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

$1$