Calculus Examples

$\frac{9 x^{2}}{x + 2}$

Step 1

Since $9$ is constant with respect to $x$ , the derivative of $\frac{9 x^{2}}{x + 2}$ with respect to $x$ is $9 \frac{d}{d x} [\frac{x^{2}}{x + 2}]$ .

$9 \frac{d}{d x} [\frac{x^{2}}{x + 2}]$

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

$9 \frac{2 (x + 2) x - x^{2} (1 + 0)}{{(x + 2)}^{2}}$

Step 3.6

Combine fractions.

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Step 3.6.1

Add $1$ and $0$ .

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

Step 3.6.2

Multiply $- 1$ by $1$ .

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

Step 3.6.3

Combine $9$ and $\frac{2 (x + 2) x - x^{2}}{{(x + 2)}^{2}}$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Simplify the numerator.

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

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

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Step 4.4.1.1.1

Move $x$ .

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

Step 4.4.1.1.2

Multiply $x$ by $x$ .

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

Step 4.4.1.2

Multiply $2$ by $9$ .

$\frac{18 x^{2} + 9 (2 \cdot 2 x) + 9 (- x^{2})}{{(x + 2)}^{2}}$

Step 4.4.1.3

Multiply $2$ by $2$ .

$\frac{18 x^{2} + 9 (4 x) + 9 (- x^{2})}{{(x + 2)}^{2}}$

Step 4.4.1.4

Multiply $4$ by $9$ .

$\frac{18 x^{2} + 36 x + 9 (- x^{2})}{{(x + 2)}^{2}}$

Step 4.4.1.5

Multiply $- 1$ by $9$ .

$\frac{18 x^{2} + 36 x - 9 x^{2}}{{(x + 2)}^{2}}$

Step 4.4.2

Subtract $9 x^{2}$ from $18 x^{2}$ .

$\frac{9 x^{2} + 36 x}{{(x + 2)}^{2}}$

Step 4.5

Factor $9 x$ out of $9 x^{2} + 36 x$ .

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Step 4.5.1

Factor $9 x$ out of $9 x^{2}$ .

$\frac{9 x (x) + 36 x}{{(x + 2)}^{2}}$

Step 4.5.2

Factor $9 x$ out of $36 x$ .

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

Step 4.5.3

Factor $9 x$ out of $9 x (x) + 9 x (4)$ .

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