Calculus Examples

$y = \frac{x^{3} + 8}{x + 2}$

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 $f (x) = x^{3} + 8$ and $g (x) = x + 2$ .

$\frac{(x + 2) \frac{d}{d x} [x^{3} + 8] - (x^{3} + 8) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{3} + 8$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [8]$ .

$\frac{(x + 2) (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [8]) - (x^{3} + 8) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{(x + 2) (3 x^{2} + \frac{d}{d x} [8]) - (x^{3} + 8) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 2.3

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

$\frac{(x + 2) (3 x^{2} + 0) - (x^{3} + 8) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $3 x^{2}$ and $0$ .

$\frac{(x + 2) (3 x^{2}) - (x^{3} + 8) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 2.4.2

Move $3$ to the left of $x + 2$ .

$\frac{3 \cdot (x + 2) x^{2} - (x^{3} + 8) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 2.5

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

$\frac{3 (x + 2) x^{2} - (x^{3} + 8) (\frac{d}{d x} [x] + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 2.6

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

$\frac{3 (x + 2) x^{2} - (x^{3} + 8) (1 + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 2.7

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{3 (x + 2) x^{2} - (x^{3} + 8) (1 + 0)}{{(x + 2)}^{2}}$

Step 2.8

Simplify the expression.

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Step 2.8.1

Add $1$ and $0$ .

$\frac{3 (x + 2) x^{2} - (x^{3} + 8) \cdot 1}{{(x + 2)}^{2}}$

Step 2.8.2

Multiply $- 1$ by $1$ .

$\frac{3 (x + 2) x^{2} - (x^{3} + 8)}{{(x + 2)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(3 x + 3 \cdot 2) x^{2} - (x^{3} + 8)}{{(x + 2)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{3 x \cdot x^{2} + 3 \cdot 2 x^{2} - (x^{3} + 8)}{{(x + 2)}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{3 x \cdot x^{2} + 3 \cdot 2 x^{2} - x^{3} - 1 \cdot 8}{{(x + 2)}^{2}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.4.1.1.1

Move $x^{2}$ .

$\frac{3 (x^{2} x) + 3 \cdot 2 x^{2} - x^{3} - 1 \cdot 8}{{(x + 2)}^{2}}$

Step 3.4.1.1.2

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

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Step 3.4.1.1.2.1

Raise $x$ to the power of $1$ .

$\frac{3 (x^{2} x^{1}) + 3 \cdot 2 x^{2} - x^{3} - 1 \cdot 8}{{(x + 2)}^{2}}$

Step 3.4.1.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 x^{2 + 1} + 3 \cdot 2 x^{2} - x^{3} - 1 \cdot 8}{{(x + 2)}^{2}}$

Step 3.4.1.1.3

Add $2$ and $1$ .

$\frac{3 x^{3} + 3 \cdot 2 x^{2} - x^{3} - 1 \cdot 8}{{(x + 2)}^{2}}$

Step 3.4.1.2

Multiply $3$ by $2$ .

$\frac{3 x^{3} + 6 x^{2} - x^{3} - 1 \cdot 8}{{(x + 2)}^{2}}$

Step 3.4.1.3

Multiply $- 1$ by $8$ .

$\frac{3 x^{3} + 6 x^{2} - x^{3} - 8}{{(x + 2)}^{2}}$

Step 3.4.2

Subtract $x^{3}$ from $3 x^{3}$ .

$\frac{2 x^{3} + 6 x^{2} - 8}{{(x + 2)}^{2}}$

Step 3.5

Factor $2$ out of $2 x^{3} + 6 x^{2} - 8$ .

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Step 3.5.1

Factor $2$ out of $2 x^{3}$ .

$\frac{2 (x^{3}) + 6 x^{2} - 8}{{(x + 2)}^{2}}$

Step 3.5.2

Factor $2$ out of $6 x^{2}$ .

$\frac{2 (x^{3}) + 2 (3 x^{2}) - 8}{{(x + 2)}^{2}}$

Step 3.5.3

Factor $2$ out of $- 8$ .

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

Step 3.5.4

Factor $2$ out of $2 x^{3} + 2 (3 x^{2})$ .

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

Step 3.5.5

Factor $2$ out of $2 (x^{3} + 3 x^{2}) + 2 \cdot - 4$ .

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