Calculus Examples

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

Step 1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 1.1

Combine $\frac{x}{4}$ and $\arcsin (x)$ .

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

Step 1.2

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

$\frac{1}{4} \frac{d}{d x} [x \arcsin (x)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \arcsin (x)$ .

$\frac{1}{4} (x \frac{d}{d x} [\arcsin (x)] + \arcsin (x) \frac{d}{d x} [x])$

Step 3

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

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

Step 4

Differentiate using the Power Rule.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Multiply $\arcsin (x)$ by $1$ .

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Apply the distributive property.

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

Step 5.2

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

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

Step 5.3

Reorder terms.

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