Calculus Examples

$f (x) = arccsc (2 x)$

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 $g (x) = 2 x$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 2

Differentiate.

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Step 2.1

Factor $2$ out of $2 x$ .

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

Step 2.2

Simplify the expression.

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Step 2.2.1

Apply the product rule to $2 (x)$ .

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

Step 2.2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify terms.

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Step 2.4.1

Multiply $2$ by $- 1$ .

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

Step 2.4.2

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

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

Step 2.4.3

Cancel the common factor of $- 2$ and $2$ .

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Step 2.4.3.1

Factor $2$ out of $- 2$ .

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

Step 2.4.3.2

Cancel the common factors.

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Step 2.4.3.2.1

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

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

Step 2.4.3.2.2

Cancel the common factor.

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

Step 2.4.3.2.3

Rewrite the expression.

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

Step 2.4.4

Move the negative in front of the fraction.

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

Step 2.5

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

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

Step 2.6

Multiply $- 1$ by $1$ .

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