Calculus Examples

$y = 5 \arcsin (2 x^{5})$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 \arcsin (2 x^{5})$ with respect to $x$ is $5 \frac{d}{d x} [\arcsin (2 x^{5})]$ .

$5 \frac{d}{d x} [\arcsin (2 x^{5})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arcsin (x)$ and $g (x) = 2 x^{5}$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $2 x^{5}$ .

$5 (\frac{d}{d u} [\arcsin (u)] \frac{d}{d x} [2 x^{5}])$

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $2 x^{5}$ .

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

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Factor $2$ out of $2 x^{5}$ .

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

Step 3.2

Combine fractions.

Tap for more steps...

Step 3.2.1

Simplify the expression.

Tap for more steps...

Step 3.2.1.1

Apply the product rule to $2 (x^{5})$ .

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

Step 3.2.1.2

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

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

Step 3.2.1.3

Multiply the exponents in ${(x^{5})}^{2}$ .

Tap for more steps...

Step 3.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.1.3.2

Multiply $5$ by $2$ .

$5 (\frac{1}{\sqrt{1 - (4 x^{10})}} \frac{d}{d x} [2 x^{5}])$

Step 3.2.1.4

Multiply $4$ by $- 1$ .

$5 (\frac{1}{\sqrt{1 - 4 x^{10}}} \frac{d}{d x} [2 x^{5}])$

Step 3.2.2

Combine $\frac{1}{\sqrt{1 - 4 x^{10}}}$ and $5$ .

$\frac{5}{\sqrt{1 - 4 x^{10}}} \frac{d}{d x} [2 x^{5}]$

Step 3.3

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{5}$ with respect to $x$ is $2 \frac{d}{d x} [x^{5}]$ .

$\frac{5}{\sqrt{1 - 4 x^{10}}} (2 \frac{d}{d x} [x^{5}])$

Step 3.4

Combine fractions.

Tap for more steps...

Step 3.4.1

Combine $2$ and $\frac{5}{\sqrt{1 - 4 x^{10}}}$ .

$\frac{2 \cdot 5}{\sqrt{1 - 4 x^{10}}} \frac{d}{d x} [x^{5}]$

Step 3.4.2

Multiply $2$ by $5$ .

$\frac{10}{\sqrt{1 - 4 x^{10}}} \frac{d}{d x} [x^{5}]$

Step 3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{10}{\sqrt{1 - 4 x^{10}}} (5 x^{4})$

Step 3.6

Combine fractions.

Tap for more steps...

Step 3.6.1

Combine $5$ and $\frac{10}{\sqrt{1 - 4 x^{10}}}$ .

$\frac{5 \cdot 10}{\sqrt{1 - 4 x^{10}}} x^{4}$

Step 3.6.2

Multiply $5$ by $10$ .

$\frac{50}{\sqrt{1 - 4 x^{10}}} x^{4}$

Step 3.6.3

Combine $\frac{50}{\sqrt{1 - 4 x^{10}}}$ and $x^{4}$ .

$\frac{50 x^{4}}{\sqrt{1 - 4 x^{10}}}$