Calculus Examples

$y = \frac{2 x - 9}{2 x^{2} + 8}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{2 x - 9}{2 x^{2} + 8})$

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 $f (x) = 2 x - 9$ and $g (x) = 2 x^{2} + 8$ .

$\frac{(2 x^{2} + 8) \frac{d}{d x} [2 x - 9] - (2 x - 9) \frac{d}{d x} [2 x^{2} + 8]}{{(2 x^{2} + 8)}^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

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

$\frac{(2 x^{2} + 8) (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 9]) - (2 x - 9) \frac{d}{d x} [2 x^{2} + 8]}{{(2 x^{2} + 8)}^{2}}$

Step 3.2.2

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

$\frac{(2 x^{2} + 8) (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]) - (2 x - 9) \frac{d}{d x} [2 x^{2} + 8]}{{(2 x^{2} + 8)}^{2}}$

Step 3.2.3

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

$\frac{(2 x^{2} + 8) (2 \cdot 1 + \frac{d}{d x} [- 9]) - (2 x - 9) \frac{d}{d x} [2 x^{2} + 8]}{{(2 x^{2} + 8)}^{2}}$

Step 3.2.4

Multiply $2$ by $1$ .

$\frac{(2 x^{2} + 8) (2 + \frac{d}{d x} [- 9]) - (2 x - 9) \frac{d}{d x} [2 x^{2} + 8]}{{(2 x^{2} + 8)}^{2}}$

Step 3.2.5

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$\frac{(2 x^{2} + 8) (2 + 0) - (2 x - 9) \frac{d}{d x} [2 x^{2} + 8]}{{(2 x^{2} + 8)}^{2}}$

Step 3.2.6

Simplify the expression.

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Step 3.2.6.1

Add $2$ and $0$ .

$\frac{(2 x^{2} + 8) \cdot 2 - (2 x - 9) \frac{d}{d x} [2 x^{2} + 8]}{{(2 x^{2} + 8)}^{2}}$

Step 3.2.6.2

Move $2$ to the left of $2 x^{2} + 8$ .

$\frac{2 \cdot (2 x^{2} + 8) - (2 x - 9) \frac{d}{d x} [2 x^{2} + 8]}{{(2 x^{2} + 8)}^{2}}$

Step 3.2.7

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

$\frac{2 (2 x^{2} + 8) - (2 x - 9) (\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [8])}{{(2 x^{2} + 8)}^{2}}$

Step 3.2.8

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

$\frac{2 (2 x^{2} + 8) - (2 x - 9) (2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [8])}{{(2 x^{2} + 8)}^{2}}$

Step 3.2.9

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

$\frac{2 (2 x^{2} + 8) - (2 x - 9) (2 (2 x) + \frac{d}{d x} [8])}{{(2 x^{2} + 8)}^{2}}$

Step 3.2.10

Multiply $2$ by $2$ .

$\frac{2 (2 x^{2} + 8) - (2 x - 9) (4 x + \frac{d}{d x} [8])}{{(2 x^{2} + 8)}^{2}}$

Step 3.2.11

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

$\frac{2 (2 x^{2} + 8) - (2 x - 9) (4 x + 0)}{{(2 x^{2} + 8)}^{2}}$

Step 3.2.12

Simplify the expression.

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Step 3.2.12.1

Add $4 x$ and $0$ .

$\frac{2 (2 x^{2} + 8) - (2 x - 9) (4 x)}{{(2 x^{2} + 8)}^{2}}$

Step 3.2.12.2

Multiply $4$ by $- 1$ .

$\frac{2 (2 x^{2} + 8) - 4 (2 x - 9) x}{{(2 x^{2} + 8)}^{2}}$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$\frac{2 (2 x^{2}) + 2 \cdot 8 - 4 (2 x - 9) x}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.2

Apply the distributive property.

$\frac{2 (2 x^{2}) + 2 \cdot 8 + (- 4 (2 x) - 4 \cdot - 9) x}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.3

Apply the distributive property.

$\frac{2 (2 x^{2}) + 2 \cdot 8 - 4 (2 x) x - 4 \cdot - 9 x}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.4

Simplify the numerator.

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Step 3.3.4.1

Simplify each term.

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Step 3.3.4.1.1

Multiply $2$ by $2$ .

$\frac{4 x^{2} + 2 \cdot 8 - 4 (2 x) x - 4 \cdot - 9 x}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.4.1.2

Multiply $2$ by $8$ .

$\frac{4 x^{2} + 16 - 4 (2 x) x - 4 \cdot - 9 x}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.4.1.3

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

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Step 3.3.4.1.3.1

Move $x$ .

$\frac{4 x^{2} + 16 - 4 (2 (x \cdot x)) - 4 \cdot - 9 x}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.4.1.3.2

Multiply $x$ by $x$ .

$\frac{4 x^{2} + 16 - 4 (2 x^{2}) - 4 \cdot - 9 x}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.4.1.4

Multiply $2$ by $- 4$ .

$\frac{4 x^{2} + 16 - 8 x^{2} - 4 \cdot - 9 x}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.4.1.5

Multiply $- 4$ by $- 9$ .

$\frac{4 x^{2} + 16 - 8 x^{2} + 36 x}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.4.2

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

$\frac{- 4 x^{2} + 16 + 36 x}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.5

Reorder terms.

$\frac{- 4 x^{2} + 36 x + 16}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.6

Factor $4$ out of $- 4 x^{2} + 36 x + 16$ .

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Step 3.3.6.1

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

$\frac{4 (- x^{2}) + 36 x + 16}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.6.2

Factor $4$ out of $36 x$ .

$\frac{4 (- x^{2}) + 4 (9 x) + 16}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.6.3

Factor $4$ out of $16$ .

$\frac{4 (- x^{2}) + 4 (9 x) + 4 (4)}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.6.4

Factor $4$ out of $4 (- x^{2}) + 4 (9 x)$ .

$\frac{4 (- x^{2} + 9 x) + 4 (4)}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.6.5

Factor $4$ out of $4 (- x^{2} + 9 x) + 4 (4)$ .

$\frac{4 (- x^{2} + 9 x + 4)}{{(2 x^{2} + 8)}^{2}}$

Step 3.3.7

Simplify the denominator.

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Step 3.3.7.1

Factor $2$ out of $2 x^{2} + 8$ .

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Step 3.3.7.1.1

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

$\frac{4 (- x^{2} + 9 x + 4)}{{(2 (x^{2}) + 8)}^{2}}$

Step 3.3.7.1.2

Factor $2$ out of $8$ .

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

Step 3.3.7.1.3

Factor $2$ out of $2 x^{2} + 2 \cdot 4$ .

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

Step 3.3.7.2

Apply the product rule to $2 (x^{2} + 4)$ .

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

Step 3.3.7.3

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

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

Step 3.3.8

Cancel the common factor of $4$ .

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Step 3.3.8.1

Cancel the common factor.

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

Step 3.3.8.2

Rewrite the expression.

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

Step 3.3.9

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

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

Step 3.3.10

Factor $- 1$ out of $9 x$ .

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

Step 3.3.11

Factor $- 1$ out of $- (x^{2}) - (- 9 x)$ .

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

Step 3.3.12

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{- (x^{2} - 9 x) - 1 (- 4)}{{(x^{2} + 4)}^{2}}$

Step 3.3.13

Factor $- 1$ out of $- (x^{2} - 9 x) - 1 (- 4)$ .

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

Step 3.3.14

Rewrite $- (x^{2} - 9 x - 4)$ as $- 1 (x^{2} - 9 x - 4)$ .

$\frac{- 1 (x^{2} - 9 x - 4)}{{(x^{2} + 4)}^{2}}$

Step 3.3.15

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

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