Calculus Examples

$y = arccsc (\frac{x}{2})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = arccsc (x)$ and $g (x) = \frac{x}{2}$ .

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Step 1.1

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

$\frac{d}{d u} [arccsc (u)] \frac{d}{d x} [\frac{x}{2}]$

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{x \sqrt{{(\frac{x}{2})}^{2} - 1}}{2}$ .

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

Step 4

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

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

Step 5

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

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

Step 6

Simplify terms.

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Step 6.1

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

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

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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

Step 7

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

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

Step 8

Multiply $- 1$ by $1$ .

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

Step 9

Simplify.

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Step 9.1

Apply the product rule to $\frac{x}{2}$ .

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

Step 9.2

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

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