Calculus Examples

$f (x) = 7 {(3 x)}^{2} - 14$

Step 1

Differentiate using the Sum Rule.

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Step 1.1

Apply the product rule to $3 x$ .

$\frac{d}{d x} [7 (3^{2} x^{2}) - 14]$

Step 1.2

Raise $3$ to the power of $2$ .

$\frac{d}{d x} [7 (9 x^{2}) - 14]$

Step 1.3

By the Sum Rule, the derivative of $7 (9 x^{2}) - 14$ with respect to $x$ is $\frac{d}{d x} [7 (9 x^{2})] + \frac{d}{d x} [- 14]$ .

$\frac{d}{d x} [7 (9 x^{2})] + \frac{d}{d x} [- 14]$

Step 2

Evaluate $\frac{d}{d x} [7 (9 x^{2})]$ .

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Step 2.1

Multiply $9$ by $7$ .

$\frac{d}{d x} [63 x^{2}] + \frac{d}{d x} [- 14]$

Step 2.2

Since $63$ is constant with respect to $x$ , the derivative of $63 x^{2}$ with respect to $x$ is $63 \frac{d}{d x} [x^{2}]$ .

$63 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 14]$

Step 2.3

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

$63 (2 x) + \frac{d}{d x} [- 14]$

Step 2.4

Multiply $2$ by $63$ .

$126 x + \frac{d}{d x} [- 14]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14$ with respect to $x$ is $0$ .

$126 x + 0$

Step 3.2

Add $126 x$ and $0$ .

$126 x$