Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Move $2$ to the left of $9 + 4 x$ .

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

Step 2.3

By the Sum Rule, the derivative of $9 + 4 x$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [4 x]$ .

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

Step 2.4

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

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

Step 2.5

Add $0$ and $\frac{d}{d x} [4 x]$ .

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

Step 2.6

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 2.7

Multiply $4$ by $- 1$ .

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

Step 2.8

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

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

Step 2.9

Multiply $- 4$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

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 $9$ .

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

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $x$ .

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

Step 3.3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.3.1.3

Multiply $4$ by $2$ .

$\frac{18 x + 8 x^{2} - 4 x^{2}}{{(9 + 4 x)}^{2}}$

Step 3.3.2

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

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

Step 3.4

Reorder terms.

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

Step 3.5

Factor $2 x$ out of $4 x^{2} + 18 x$ .

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Step 3.5.1

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

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

Step 3.5.2

Factor $2 x$ out of $18 x$ .

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

Step 3.5.3

Factor $2 x$ out of $2 x (2 x) + 2 x (9)$ .

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