Calculus Examples

$f (x) = - 2 arcsec (x^{2})$

Step 1

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

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

Step 2

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 $g (x) = x^{2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.1.2

Multiply $2$ by $2$ .

$- 2 (\frac{1}{x^{2} \sqrt{x^{4} - 1}} \frac{d}{d x} [x^{2}])$

Step 3.2

Combine fractions.

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Step 3.2.1

Combine $\frac{1}{x^{2} \sqrt{x^{4} - 1}}$ and $- 2$ .

$\frac{- 2}{x^{2} \sqrt{x^{4} - 1}} \frac{d}{d x} [x^{2}]$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{2}{x^{2} \sqrt{x^{4} - 1}} \frac{d}{d x} [x^{2}]$

Step 3.3

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

$- \frac{2}{x^{2} \sqrt{x^{4} - 1}} (2 x)$

Step 3.4

Simplify terms.

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Step 3.4.1

Multiply $2$ by $- 1$ .

$- 2 \frac{2}{x^{2} \sqrt{x^{4} - 1}} x$

Step 3.4.2

Combine $- 2$ and $\frac{2}{x^{2} \sqrt{x^{4} - 1}}$ .

$\frac{- 2 \cdot 2}{x^{2} \sqrt{x^{4} - 1}} x$

Step 3.4.3

Multiply $- 2$ by $2$ .

$\frac{- 4}{x^{2} \sqrt{x^{4} - 1}} x$

Step 3.4.4

Combine $\frac{- 4}{x^{2} \sqrt{x^{4} - 1}}$ and $x$ .

$\frac{- 4 x}{x^{2} \sqrt{x^{4} - 1}}$

Step 3.4.5

Cancel the common factor of $x$ and $x^{2}$ .

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Step 3.4.5.1

Factor $x$ out of $- 4 x$ .

$\frac{x \cdot - 4}{x^{2} \sqrt{x^{4} - 1}}$

Step 3.4.5.2

Cancel the common factors.

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Step 3.4.5.2.1

Factor $x$ out of $x^{2} \sqrt{x^{4} - 1}$ .

$\frac{x \cdot - 4}{x (x \sqrt{x^{4} - 1})}$

Step 3.4.5.2.2

Cancel the common factor.

$\frac{x \cdot - 4}{x (x \sqrt{x^{4} - 1})}$

Step 3.4.5.2.3

Rewrite the expression.

$\frac{- 4}{x \sqrt{x^{4} - 1}}$

Step 3.4.6

Move the negative in front of the fraction.

$- \frac{4}{x \sqrt{x^{4} - 1}}$