Calculus Examples

$g (x) = 5 \arccos (\frac{x}{4})$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 \arccos (\frac{x}{4})$ with respect to $x$ is $5 \frac{d}{d x} [\arccos (\frac{x}{4})]$ .

$5 \frac{d}{d x} [\arccos (\frac{x}{4})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arccos (x)$ and $g (x) = \frac{x}{4}$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $\frac{x}{4}$ .

$5 (\frac{d}{d u} [\arccos (u)] \frac{d}{d x} [\frac{x}{4}])$

Step 2.2

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

$5 (- \frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [\frac{x}{4}])$

Step 2.3

Replace all occurrences of $u$ with $\frac{x}{4}$ .

$5 (- \frac{1}{\sqrt{1 - {(\frac{x}{4})}^{2}}} \frac{d}{d x} [\frac{x}{4}])$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Multiply $- 1$ by $5$ .

$- 5 (\frac{1}{\sqrt{1 - {(\frac{x}{4})}^{2}}} \frac{d}{d x} [\frac{x}{4}])$

Step 3.2

Combine fractions.

Tap for more steps...

Step 3.2.1

Combine $\frac{1}{\sqrt{1 - {(\frac{x}{4})}^{2}}}$ and $- 5$ .

$\frac{- 5}{\sqrt{1 - {(\frac{x}{4})}^{2}}} \frac{d}{d x} [\frac{x}{4}]$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{5}{\sqrt{1 - {(\frac{x}{4})}^{2}}} \frac{d}{d x} [\frac{x}{4}]$

Step 3.3

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x}{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x]$ .

$- \frac{5}{\sqrt{1 - {(\frac{x}{4})}^{2}}} (\frac{1}{4} \frac{d}{d x} [x])$

Step 3.4

Multiply $\frac{1}{4}$ by $\frac{5}{\sqrt{1 - {(\frac{x}{4})}^{2}}}$ .

$- \frac{5}{4 \sqrt{1 - {(\frac{x}{4})}^{2}}} \frac{d}{d x} [x]$

Step 3.5

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

$- \frac{5}{4 \sqrt{1 - {(\frac{x}{4})}^{2}}} \cdot 1$

Step 3.6

Multiply $- 1$ by $1$ .

$- \frac{5}{4 \sqrt{1 - {(\frac{x}{4})}^{2}}}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the product rule to $\frac{x}{4}$ .

$- \frac{5}{4 \sqrt{1 - \frac{x^{2}}{4^{2}}}}$

Step 4.2

Raise $4$ to the power of $2$ .

$- \frac{5}{4 \sqrt{1 - \frac{x^{2}}{16}}}$