Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $2 x$ and $0$ .

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

Step 2.4.2

Move $2$ to the left of $8 - x$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 2.8

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

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

Step 2.9

Multiply.

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Step 2.9.1

Multiply $- 1$ by $- 1$ .

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

Step 2.9.2

Multiply $x^{2} + 8$ by $1$ .

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

Step 2.10

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

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

Step 2.11

Multiply $x^{2} + 8$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $2$ by $8$ .

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

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $x$ .

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

Step 3.3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.3.1.3

Multiply $- 1$ by $2$ .

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

Step 3.3.2

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

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

Step 3.4

Reorder terms.

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

Step 3.5

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

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

Step 3.6

Factor $- 1$ out of $16 x$ .

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

Step 3.7

Factor $- 1$ out of $- (x^{2}) - (- 16 x)$ .

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

Step 3.8

Rewrite $8$ as $- 1 (- 8)$ .

$\frac{- (x^{2} - 16 x) - 1 (- 8)}{{(- x + 8)}^{2}}$

Step 3.9

Factor $- 1$ out of $- (x^{2} - 16 x) - 1 (- 8)$ .

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

Step 3.10

Rewrite $- (x^{2} - 16 x - 8)$ as $- 1 (x^{2} - 16 x - 8)$ .

$\frac{- 1 (x^{2} - 16 x - 8)}{{(- x + 8)}^{2}}$

Step 3.11

Move the negative in front of the fraction.

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