Calculus Examples

$y = \arcsin (\frac{x}{4})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\arcsin (\frac{x}{4}))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

Differentiate.

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.3

Simplify.

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

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

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{1}{4 \sqrt{1 - \frac{x^{2}}{16}}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{1}{4 \sqrt{1 - \frac{x^{2}}{16}}}$