Calculus Examples

$\arccos (\frac{x}{2})$

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $- 1$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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

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