Calculus Examples

$y = \frac{5 x^{2} - 9 x + 13}{2 x + 4}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{5 x^{2} - 9 x + 13}{2 x + 4})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = 5 x^{2} - 9 x + 13$ and $g (x) = 2 x + 4$ .

$\frac{(2 x + 4) \frac{d}{d x} [5 x^{2} - 9 x + 13] - (5 x^{2} - 9 x + 13) \frac{d}{d x} [2 x + 4]}{{(2 x + 4)}^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $5 x^{2} - 9 x + 13$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [13]$ .

$\frac{(2 x + 4) (\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [13]) - (5 x^{2} - 9 x + 13) \frac{d}{d x} [2 x + 4]}{{(2 x + 4)}^{2}}$

Step 3.2.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$\frac{(2 x + 4) (5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [13]) - (5 x^{2} - 9 x + 13) \frac{d}{d x} [2 x + 4]}{{(2 x + 4)}^{2}}$

Step 3.2.3

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

$\frac{(2 x + 4) (5 (2 x) + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [13]) - (5 x^{2} - 9 x + 13) \frac{d}{d x} [2 x + 4]}{{(2 x + 4)}^{2}}$

Step 3.2.4

Multiply $2$ by $5$ .

$\frac{(2 x + 4) (10 x + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [13]) - (5 x^{2} - 9 x + 13) \frac{d}{d x} [2 x + 4]}{{(2 x + 4)}^{2}}$

Step 3.2.5

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$\frac{(2 x + 4) (10 x - 9 \frac{d}{d x} [x] + \frac{d}{d x} [13]) - (5 x^{2} - 9 x + 13) \frac{d}{d x} [2 x + 4]}{{(2 x + 4)}^{2}}$

Step 3.2.6

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

$\frac{(2 x + 4) (10 x - 9 \cdot 1 + \frac{d}{d x} [13]) - (5 x^{2} - 9 x + 13) \frac{d}{d x} [2 x + 4]}{{(2 x + 4)}^{2}}$

Step 3.2.7

Multiply $- 9$ by $1$ .

$\frac{(2 x + 4) (10 x - 9 + \frac{d}{d x} [13]) - (5 x^{2} - 9 x + 13) \frac{d}{d x} [2 x + 4]}{{(2 x + 4)}^{2}}$

Step 3.2.8

Since $13$ is constant with respect to $x$ , the derivative of $13$ with respect to $x$ is $0$ .

$\frac{(2 x + 4) (10 x - 9 + 0) - (5 x^{2} - 9 x + 13) \frac{d}{d x} [2 x + 4]}{{(2 x + 4)}^{2}}$

Step 3.2.9

Add $10 x - 9$ and $0$ .

$\frac{(2 x + 4) (10 x - 9) - (5 x^{2} - 9 x + 13) \frac{d}{d x} [2 x + 4]}{{(2 x + 4)}^{2}}$

Step 3.2.10

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

$\frac{(2 x + 4) (10 x - 9) - (5 x^{2} - 9 x + 13) (\frac{d}{d x} [2 x] + \frac{d}{d x} [4])}{{(2 x + 4)}^{2}}$

Step 3.2.11

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

$\frac{(2 x + 4) (10 x - 9) - (5 x^{2} - 9 x + 13) (2 \frac{d}{d x} [x] + \frac{d}{d x} [4])}{{(2 x + 4)}^{2}}$

Step 3.2.12

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

$\frac{(2 x + 4) (10 x - 9) - (5 x^{2} - 9 x + 13) (2 \cdot 1 + \frac{d}{d x} [4])}{{(2 x + 4)}^{2}}$

Step 3.2.13

Multiply $2$ by $1$ .

$\frac{(2 x + 4) (10 x - 9) - (5 x^{2} - 9 x + 13) (2 + \frac{d}{d x} [4])}{{(2 x + 4)}^{2}}$

Step 3.2.14

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

$\frac{(2 x + 4) (10 x - 9) - (5 x^{2} - 9 x + 13) (2 + 0)}{{(2 x + 4)}^{2}}$

Step 3.2.15

Simplify the expression.

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Step 3.2.15.1

Add $2$ and $0$ .

$\frac{(2 x + 4) (10 x - 9) - (5 x^{2} - 9 x + 13) \cdot 2}{{(2 x + 4)}^{2}}$

Step 3.2.15.2

Multiply $2$ by $- 1$ .

$\frac{(2 x + 4) (10 x - 9) - 2 (5 x^{2} - 9 x + 13)}{{(2 x + 4)}^{2}}$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$\frac{(2 x + 4) (10 x - 9) - 2 (5 x^{2}) - 2 (- 9 x) - 2 \cdot 13}{{(2 x + 4)}^{2}}$

Step 3.3.2

Simplify the numerator.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Expand $(2 x + 4) (10 x - 9)$ using the FOIL Method.

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Step 3.3.2.1.1.1

Apply the distributive property.

$\frac{2 x (10 x - 9) + 4 (10 x - 9) - 2 (5 x^{2}) - 2 (- 9 x) - 2 \cdot 13}{{(2 x + 4)}^{2}}$

Step 3.3.2.1.1.2

Apply the distributive property.

$\frac{2 x (10 x) + 2 x \cdot - 9 + 4 (10 x - 9) - 2 (5 x^{2}) - 2 (- 9 x) - 2 \cdot 13}{{(2 x + 4)}^{2}}$

Step 3.3.2.1.1.3

Apply the distributive property.

$\frac{2 x (10 x) + 2 x \cdot - 9 + 4 (10 x) + 4 \cdot - 9 - 2 (5 x^{2}) - 2 (- 9 x) - 2 \cdot 13}{{(2 x + 4)}^{2}}$

Step 3.3.2.1.2

Simplify and combine like terms.

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Step 3.3.2.1.2.1

Simplify each term.

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Step 3.3.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 \cdot 10 x \cdot x + 2 x \cdot - 9 + 4 (10 x) + 4 \cdot - 9 - 2 (5 x^{2}) - 2 (- 9 x) - 2 \cdot 13}{{(2 x + 4)}^{2}}$

Step 3.3.2.1.2.1.2

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

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Step 3.3.2.1.2.1.2.1

Move $x$ .

$\frac{2 \cdot 10 (x \cdot x) + 2 x \cdot - 9 + 4 (10 x) + 4 \cdot - 9 - 2 (5 x^{2}) - 2 (- 9 x) - 2 \cdot 13}{{(2 x + 4)}^{2}}$

Step 3.3.2.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{2 \cdot 10 x^{2} + 2 x \cdot - 9 + 4 (10 x) + 4 \cdot - 9 - 2 (5 x^{2}) - 2 (- 9 x) - 2 \cdot 13}{{(2 x + 4)}^{2}}$

Step 3.3.2.1.2.1.3

Multiply $2$ by $10$ .

$\frac{20 x^{2} + 2 x \cdot - 9 + 4 (10 x) + 4 \cdot - 9 - 2 (5 x^{2}) - 2 (- 9 x) - 2 \cdot 13}{{(2 x + 4)}^{2}}$

Step 3.3.2.1.2.1.4

Multiply $- 9$ by $2$ .

$\frac{20 x^{2} - 18 x + 4 (10 x) + 4 \cdot - 9 - 2 (5 x^{2}) - 2 (- 9 x) - 2 \cdot 13}{{(2 x + 4)}^{2}}$

Step 3.3.2.1.2.1.5

Multiply $10$ by $4$ .

$\frac{20 x^{2} - 18 x + 40 x + 4 \cdot - 9 - 2 (5 x^{2}) - 2 (- 9 x) - 2 \cdot 13}{{(2 x + 4)}^{2}}$

Step 3.3.2.1.2.1.6

Multiply $4$ by $- 9$ .

$\frac{20 x^{2} - 18 x + 40 x - 36 - 2 (5 x^{2}) - 2 (- 9 x) - 2 \cdot 13}{{(2 x + 4)}^{2}}$

Step 3.3.2.1.2.2

Add $- 18 x$ and $40 x$ .

$\frac{20 x^{2} + 22 x - 36 - 2 (5 x^{2}) - 2 (- 9 x) - 2 \cdot 13}{{(2 x + 4)}^{2}}$

Step 3.3.2.1.3

Multiply $5$ by $- 2$ .

$\frac{20 x^{2} + 22 x - 36 - 10 x^{2} - 2 (- 9 x) - 2 \cdot 13}{{(2 x + 4)}^{2}}$

Step 3.3.2.1.4

Multiply $- 9$ by $- 2$ .

$\frac{20 x^{2} + 22 x - 36 - 10 x^{2} + 18 x - 2 \cdot 13}{{(2 x + 4)}^{2}}$

Step 3.3.2.1.5

Multiply $- 2$ by $13$ .

$\frac{20 x^{2} + 22 x - 36 - 10 x^{2} + 18 x - 26}{{(2 x + 4)}^{2}}$

Step 3.3.2.2

Subtract $10 x^{2}$ from $20 x^{2}$ .

$\frac{10 x^{2} + 22 x - 36 + 18 x - 26}{{(2 x + 4)}^{2}}$

Step 3.3.2.3

Add $22 x$ and $18 x$ .

$\frac{10 x^{2} + 40 x - 36 - 26}{{(2 x + 4)}^{2}}$

Step 3.3.2.4

Subtract $26$ from $- 36$ .

$\frac{10 x^{2} + 40 x - 62}{{(2 x + 4)}^{2}}$

Step 3.3.3

Factor $2$ out of $10 x^{2} + 40 x - 62$ .

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Step 3.3.3.1

Factor $2$ out of $10 x^{2}$ .

$\frac{2 (5 x^{2}) + 40 x - 62}{{(2 x + 4)}^{2}}$

Step 3.3.3.2

Factor $2$ out of $40 x$ .

$\frac{2 (5 x^{2}) + 2 (20 x) - 62}{{(2 x + 4)}^{2}}$

Step 3.3.3.3

Factor $2$ out of $- 62$ .

$\frac{2 (5 x^{2}) + 2 (20 x) + 2 (- 31)}{{(2 x + 4)}^{2}}$

Step 3.3.3.4

Factor $2$ out of $2 (5 x^{2}) + 2 (20 x)$ .

$\frac{2 (5 x^{2} + 20 x) + 2 (- 31)}{{(2 x + 4)}^{2}}$

Step 3.3.3.5

Factor $2$ out of $2 (5 x^{2} + 20 x) + 2 (- 31)$ .

$\frac{2 (5 x^{2} + 20 x - 31)}{{(2 x + 4)}^{2}}$

Step 3.3.4

Simplify the denominator.

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Step 3.3.4.1

Factor $2$ out of $2 x + 4$ .

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Step 3.3.4.1.1

Factor $2$ out of $2 x$ .

$\frac{2 (5 x^{2} + 20 x - 31)}{{(2 (x) + 4)}^{2}}$

Step 3.3.4.1.2

Factor $2$ out of $4$ .

$\frac{2 (5 x^{2} + 20 x - 31)}{{(2 x + 2 \cdot 2)}^{2}}$

Step 3.3.4.1.3

Factor $2$ out of $2 x + 2 \cdot 2$ .

$\frac{2 (5 x^{2} + 20 x - 31)}{{(2 (x + 2))}^{2}}$

Step 3.3.4.2

Apply the product rule to $2 (x + 2)$ .

$\frac{2 (5 x^{2} + 20 x - 31)}{2^{2} {(x + 2)}^{2}}$

Step 3.3.4.3

Raise $2$ to the power of $2$ .

$\frac{2 (5 x^{2} + 20 x - 31)}{4 {(x + 2)}^{2}}$

Step 3.3.5

Cancel the common factors.

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Step 3.3.5.1

Factor $2$ out of $4 {(x + 2)}^{2}$ .

$\frac{2 (5 x^{2} + 20 x - 31)}{2 (2 {(x + 2)}^{2})}$

Step 3.3.5.2

Cancel the common factor.

$\frac{2 (5 x^{2} + 20 x - 31)}{2 (2 {(x + 2)}^{2})}$

Step 3.3.5.3

Rewrite the expression.

$\frac{5 x^{2} + 20 x - 31}{2 {(x + 2)}^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{5 x^{2} + 20 x - 31}{2 {(x + 2)}^{2}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{5 x^{2} + 20 x - 31}{2 {(x + 2)}^{2}}$