Calculus Examples

$g (x) = \frac{9 (7 x^{2} + 5 x)}{4 x}$

Step 1

Since $\frac{9}{4}$ is constant with respect to $x$ , the derivative of $\frac{9 (7 x^{2} + 5 x)}{4 x}$ with respect to $x$ is $\frac{9}{4} \frac{d}{d x} [\frac{7 x^{2} + 5 x}{x}]$ .

$\frac{9}{4} \frac{d}{d x} [\frac{7 x^{2} + 5 x}{x}]$

Step 2

Simplify terms.

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Step 2.1

Factor $x$ out of $7 x^{2} + 5 x$ .

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Step 2.1.1

Factor $x$ out of $7 x^{2}$ .

$\frac{9}{4} \frac{d}{d x} [\frac{x (7 x) + 5 x}{x}]$

Step 2.1.2

Factor $x$ out of $5 x$ .

$\frac{9}{4} \frac{d}{d x} [\frac{x (7 x) + x \cdot 5}{x}]$

Step 2.1.3

Factor $x$ out of $x (7 x) + x \cdot 5$ .

$\frac{9}{4} \frac{d}{d x} [\frac{x (7 x + 5)}{x}]$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$\frac{9}{4} \frac{d}{d x} [\frac{x (7 x + 5)}{x}]$

Step 2.2.2

Divide $7 x + 5$ by $1$ .

$\frac{9}{4} \frac{d}{d x} [7 x + 5]$

Step 3

By the Sum Rule, the derivative of $7 x + 5$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [5]$ .

$\frac{9}{4} (\frac{d}{d x} [7 x] + \frac{d}{d x} [5])$

Step 4

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$\frac{9}{4} (7 \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 5

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

$\frac{9}{4} (7 \cdot 1 + \frac{d}{d x} [5])$

Step 6

Multiply $7$ by $1$ .

$\frac{9}{4} (7 + \frac{d}{d x} [5])$

Step 7

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{9}{4} (7 + 0)$

Step 8

Combine fractions.

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Step 8.1

Add $7$ and $0$ .

$\frac{9}{4} \cdot 7$

Step 8.2

Combine $\frac{9}{4}$ and $7$ .

$\frac{9 \cdot 7}{4}$

Step 8.3

Multiply $9$ by $7$ .

$\frac{63}{4}$