Calculus Examples

$y = \frac{- x^{2} - 4 x - 7}{x + 3}$

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 $f (x) = - x^{2} - 4 x - 7$ and $g (x) = x + 3$ .

$\frac{(x + 3) \frac{d}{d x} [- x^{2} - 4 x - 7] - (- x^{2} - 4 x - 7) \frac{d}{d x} [x + 3]}{{(x + 3)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $- x^{2} - 4 x - 7$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [- 7]$ .

$\frac{(x + 3) (\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [- 7]) - (- x^{2} - 4 x - 7) \frac{d}{d x} [x + 3]}{{(x + 3)}^{2}}$

Step 2.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$\frac{(x + 3) (- \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [- 7]) - (- x^{2} - 4 x - 7) \frac{d}{d x} [x + 3]}{{(x + 3)}^{2}}$

Step 2.3

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

$\frac{(x + 3) (- (2 x) + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [- 7]) - (- x^{2} - 4 x - 7) \frac{d}{d x} [x + 3]}{{(x + 3)}^{2}}$

Step 2.4

Multiply $2$ by $- 1$ .

$\frac{(x + 3) (- 2 x + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [- 7]) - (- x^{2} - 4 x - 7) \frac{d}{d x} [x + 3]}{{(x + 3)}^{2}}$

Step 2.5

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

$\frac{(x + 3) (- 2 x - 4 \frac{d}{d x} [x] + \frac{d}{d x} [- 7]) - (- x^{2} - 4 x - 7) \frac{d}{d x} [x + 3]}{{(x + 3)}^{2}}$

Step 2.6

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

$\frac{(x + 3) (- 2 x - 4 \cdot 1 + \frac{d}{d x} [- 7]) - (- x^{2} - 4 x - 7) \frac{d}{d x} [x + 3]}{{(x + 3)}^{2}}$

Step 2.7

Multiply $- 4$ by $1$ .

$\frac{(x + 3) (- 2 x - 4 + \frac{d}{d x} [- 7]) - (- x^{2} - 4 x - 7) \frac{d}{d x} [x + 3]}{{(x + 3)}^{2}}$

Step 2.8

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

$\frac{(x + 3) (- 2 x - 4 + 0) - (- x^{2} - 4 x - 7) \frac{d}{d x} [x + 3]}{{(x + 3)}^{2}}$

Step 2.9

Add $- 2 x - 4$ and $0$ .

$\frac{(x + 3) (- 2 x - 4) - (- x^{2} - 4 x - 7) \frac{d}{d x} [x + 3]}{{(x + 3)}^{2}}$

Step 2.10

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

$\frac{(x + 3) (- 2 x - 4) - (- x^{2} - 4 x - 7) (\frac{d}{d x} [x] + \frac{d}{d x} [3])}{{(x + 3)}^{2}}$

Step 2.11

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

$\frac{(x + 3) (- 2 x - 4) - (- x^{2} - 4 x - 7) (1 + \frac{d}{d x} [3])}{{(x + 3)}^{2}}$

Step 2.12

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

$\frac{(x + 3) (- 2 x - 4) - (- x^{2} - 4 x - 7) (1 + 0)}{{(x + 3)}^{2}}$

Step 2.13

Simplify the expression.

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Step 2.13.1

Add $1$ and $0$ .

$\frac{(x + 3) (- 2 x - 4) - (- x^{2} - 4 x - 7) \cdot 1}{{(x + 3)}^{2}}$

Step 2.13.2

Multiply $- 1$ by $1$ .

$\frac{(x + 3) (- 2 x - 4) - (- x^{2} - 4 x - 7)}{{(x + 3)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(x + 3) (- 2 x - 4) - - x^{2} - (- 4 x) - - 7}{{(x + 3)}^{2}}$

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 $(x + 3) (- 2 x - 4)$ using the FOIL Method.

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Step 3.2.1.1.1

Apply the distributive property.

$\frac{x (- 2 x - 4) + 3 (- 2 x - 4) - - x^{2} - (- 4 x) - - 7}{{(x + 3)}^{2}}$

Step 3.2.1.1.2

Apply the distributive property.

$\frac{x (- 2 x) + x \cdot - 4 + 3 (- 2 x - 4) - - x^{2} - (- 4 x) - - 7}{{(x + 3)}^{2}}$

Step 3.2.1.1.3

Apply the distributive property.

$\frac{x (- 2 x) + x \cdot - 4 + 3 (- 2 x) + 3 \cdot - 4 - - x^{2} - (- 4 x) - - 7}{{(x + 3)}^{2}}$

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.

$\frac{- 2 x \cdot x + x \cdot - 4 + 3 (- 2 x) + 3 \cdot - 4 - - x^{2} - (- 4 x) - - 7}{{(x + 3)}^{2}}$

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 $x$ .

$\frac{- 2 (x \cdot x) + x \cdot - 4 + 3 (- 2 x) + 3 \cdot - 4 - - x^{2} - (- 4 x) - - 7}{{(x + 3)}^{2}}$

Step 3.2.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{- 2 x^{2} + x \cdot - 4 + 3 (- 2 x) + 3 \cdot - 4 - - x^{2} - (- 4 x) - - 7}{{(x + 3)}^{2}}$

Step 3.2.1.2.1.3

Move $- 4$ to the left of $x$ .

$\frac{- 2 x^{2} - 4 \cdot x + 3 (- 2 x) + 3 \cdot - 4 - - x^{2} - (- 4 x) - - 7}{{(x + 3)}^{2}}$

Step 3.2.1.2.1.4

Multiply $- 2$ by $3$ .

$\frac{- 2 x^{2} - 4 x - 6 x + 3 \cdot - 4 - - x^{2} - (- 4 x) - - 7}{{(x + 3)}^{2}}$

Step 3.2.1.2.1.5

Multiply $3$ by $- 4$ .

$\frac{- 2 x^{2} - 4 x - 6 x - 12 - - x^{2} - (- 4 x) - - 7}{{(x + 3)}^{2}}$

Step 3.2.1.2.2

Subtract $6 x$ from $- 4 x$ .

$\frac{- 2 x^{2} - 10 x - 12 - - x^{2} - (- 4 x) - - 7}{{(x + 3)}^{2}}$

Step 3.2.1.3

Multiply $- - x^{2}$ .

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Step 3.2.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{- 2 x^{2} - 10 x - 12 + 1 x^{2} - (- 4 x) - - 7}{{(x + 3)}^{2}}$

Step 3.2.1.3.2

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

$\frac{- 2 x^{2} - 10 x - 12 + x^{2} - (- 4 x) - - 7}{{(x + 3)}^{2}}$

Step 3.2.1.4

Multiply $- 4$ by $- 1$ .

$\frac{- 2 x^{2} - 10 x - 12 + x^{2} + 4 x - - 7}{{(x + 3)}^{2}}$

Step 3.2.1.5

Multiply $- 1$ by $- 7$ .

$\frac{- 2 x^{2} - 10 x - 12 + x^{2} + 4 x + 7}{{(x + 3)}^{2}}$

Step 3.2.2

Add $- 2 x^{2}$ and $x^{2}$ .

$\frac{- x^{2} - 10 x - 12 + 4 x + 7}{{(x + 3)}^{2}}$

Step 3.2.3

Add $- 10 x$ and $4 x$ .

$\frac{- x^{2} - 6 x - 12 + 7}{{(x + 3)}^{2}}$

Step 3.2.4

Add $- 12$ and $7$ .

$\frac{- x^{2} - 6 x - 5}{{(x + 3)}^{2}}$

Step 3.3

Factor by grouping.

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Step 3.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 5 = 5$ and whose sum is $b = - 6$ .

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Step 3.3.1.1

Factor $- 6$ out of $- 6 x$ .

$\frac{- x^{2} - 6 (x) - 5}{{(x + 3)}^{2}}$

Step 3.3.1.2

Rewrite $- 6$ as $- 1$ plus $- 5$

$\frac{- x^{2} + (- 1 - 5) x - 5}{{(x + 3)}^{2}}$

Step 3.3.1.3

Apply the distributive property.

$\frac{- x^{2} - 1 x - 5 x - 5}{{(x + 3)}^{2}}$

Step 3.3.2

Factor out the greatest common factor from each group.

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Step 3.3.2.1

Group the first two terms and the last two terms.

$\frac{(- x^{2} - 1 x) - 5 x - 5}{{(x + 3)}^{2}}$

Step 3.3.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (- x - 1) + 5 (- x - 1)}{{(x + 3)}^{2}}$

Step 3.3.3

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

$\frac{(- x - 1) (x + 5)}{{(x + 3)}^{2}}$

Step 3.4

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

$\frac{(- (x) - 1) (x + 5)}{{(x + 3)}^{2}}$

Step 3.5

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

$\frac{(- (x) - 1 (1)) (x + 5)}{{(x + 3)}^{2}}$

Step 3.6

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

$\frac{- (x + 1) (x + 5)}{{(x + 3)}^{2}}$

Step 3.7

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

$\frac{- 1 (x + 1) (x + 5)}{{(x + 3)}^{2}}$

Step 3.8

Move the negative in front of the fraction.

$- \frac{(x + 1) (x + 5)}{{(x + 3)}^{2}}$