Calculus Examples

$arcsec (3 x)$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = arcsec (x)$ and $g (x) = 3 x$ .

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To apply the Chain Rule, set $u$ as $3 x$ .

$\frac{d}{d u} [arcsec (u)] \frac{d}{d x} [3 x]$

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

$\frac{1}{u \sqrt{u^{2} - 1}} \frac{d}{d x} [3 x]$

Replace all occurrences of $u$ with $3 x$ .

$\frac{1}{3 x \sqrt{{(3 x)}^{2} - 1}} \frac{d}{d x} [3 x]$

Differentiate.

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Factor $3$ out of $3 x$ .

$\frac{1}{3 x \sqrt{{(3 (x))}^{2} - 1}} \frac{d}{d x} [3 x]$

Simplify the expression.

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Apply the product rule to $3 (x)$ .

$\frac{1}{3 x \sqrt{3^{2} x^{2} - 1}} \frac{d}{d x} [3 x]$

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

$\frac{1}{3 x \sqrt{9 x^{2} - 1}} \frac{d}{d x} [3 x]$

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

$\frac{1}{3 x \sqrt{9 x^{2} - 1}} (3 \frac{d}{d x} [x])$

Simplify terms.

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Combine $3$ and $\frac{1}{3 x \sqrt{9 x^{2} - 1}}$ .

$\frac{3}{3 x \sqrt{9 x^{2} - 1}} \frac{d}{d x} [x]$

Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3}{3 x \sqrt{9 x^{2} - 1}} \frac{d}{d x} [x]$

Rewrite the expression.

$\frac{1}{x \sqrt{9 x^{2} - 1}} \frac{d}{d x} [x]$

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

$\frac{1}{x \sqrt{9 x^{2} - 1}} \cdot 1$

Multiply $\frac{1}{x \sqrt{9 x^{2} - 1}}$ by $1$ .

$\frac{1}{x \sqrt{9 x^{2} - 1}}$