Calculus Examples

$y = 7 x^{3} - arcsec (x)$

Step 1

By the Sum Rule, the derivative of $7 x^{3} - arcsec (x)$ with respect to $x$ is $\frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [- arcsec (x)]$ .

$\frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [- arcsec (x)]$

Step 2

Evaluate $\frac{d}{d x} [7 x^{3}]$ .

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Step 2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{3}$ with respect to $x$ is $7 \frac{d}{d x} [x^{3}]$ .

$7 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- arcsec (x)]$

Step 2.2

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

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

Step 2.3

Multiply $3$ by $7$ .

$21 x^{2} + \frac{d}{d x} [- arcsec (x)]$

Step 3

Evaluate $\frac{d}{d x} [- arcsec (x)]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- arcsec (x)$ with respect to $x$ is $- \frac{d}{d x} [arcsec (x)]$ .

$21 x^{2} - \frac{d}{d x} [arcsec (x)]$

Step 3.2

The derivative of $arcsec (x)$ with respect to $x$ is $\frac{1}{x \sqrt{x^{2} - 1}}$ .

$21 x^{2} - \frac{1}{x \sqrt{x^{2} - 1}}$