Calculus Examples

$y = \arcsin (\frac{3}{x^{2}})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arcsin (x)$ and $g (x) = \frac{3}{x^{2}}$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $\frac{3}{x^{2}}$ .

$\frac{d}{d u} [\arcsin (u)] \frac{d}{d x} [\frac{3}{x^{2}}]$

Step 1.2

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

$\frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [\frac{3}{x^{2}}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{3}{x^{2}}$ .

$\frac{1}{\sqrt{1 - {(\frac{3}{x^{2}})}^{2}}} \frac{d}{d x} [\frac{3}{x^{2}}]$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x^{2}}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$\frac{1}{\sqrt{1 - {(\frac{3}{x^{2}})}^{2}}} (3 \frac{d}{d x} [\frac{1}{x^{2}}])$

Step 2.2

Combine fractions.

Tap for more steps...

Step 2.2.1

Combine $3$ and $\frac{1}{\sqrt{1 - {(\frac{3}{x^{2}})}^{2}}}$ .

$\frac{3}{\sqrt{1 - {(\frac{3}{x^{2}})}^{2}}} \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 2.2.2

Apply basic rules of exponents.

Tap for more steps...

Step 2.2.2.1

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$\frac{3}{\sqrt{1 - {(\frac{3}{x^{2}})}^{2}}} \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 2.2.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

Tap for more steps...

Step 2.2.2.2.1

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

$\frac{3}{\sqrt{1 - {(\frac{3}{x^{2}})}^{2}}} \frac{d}{d x} [x^{2 \cdot - 1}]$

Step 2.2.2.2.2

Multiply $2$ by $- 1$ .

$\frac{3}{\sqrt{1 - {(\frac{3}{x^{2}})}^{2}}} \frac{d}{d x} [x^{- 2}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 2$ .

$\frac{3}{\sqrt{1 - {(\frac{3}{x^{2}})}^{2}}} (- 2 x^{- 3})$

Step 2.4

Combine fractions.

Tap for more steps...

Step 2.4.1

Combine $- 2$ and $\frac{3}{\sqrt{1 - {(\frac{3}{x^{2}})}^{2}}}$ .

$\frac{- 2 \cdot 3}{\sqrt{1 - {(\frac{3}{x^{2}})}^{2}}} x^{- 3}$

Step 2.4.2

Multiply $- 2$ by $3$ .

$\frac{- 6}{\sqrt{1 - {(\frac{3}{x^{2}})}^{2}}} x^{- 3}$

Step 2.4.3

Combine $\frac{- 6}{\sqrt{1 - {(\frac{3}{x^{2}})}^{2}}}$ and $x^{- 3}$ .

$\frac{- 6 x^{- 3}}{\sqrt{1 - {(\frac{3}{x^{2}})}^{2}}}$

Step 2.4.4

Simplify the expression.

Tap for more steps...

Step 2.4.4.1

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 6}{\sqrt{1 - {(\frac{3}{x^{2}})}^{2}} x^{3}}$

Step 2.4.4.2

Move the negative in front of the fraction.

$- \frac{6}{\sqrt{1 - {(\frac{3}{x^{2}})}^{2}} x^{3}}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the product rule to $\frac{3}{x^{2}}$ .

$- \frac{6}{\sqrt{1 - \frac{3^{2}}{{(x^{2})}^{2}}} x^{3}}$

Step 3.2

Combine terms.

Tap for more steps...

Step 3.2.1

Raise $3$ to the power of $2$ .

$- \frac{6}{\sqrt{1 - \frac{9}{{(x^{2})}^{2}}} x^{3}}$

Step 3.2.2

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

Tap for more steps...

Step 3.2.2.1

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

$- \frac{6}{\sqrt{1 - \frac{9}{x^{2 \cdot 2}}} x^{3}}$

Step 3.2.2.2

Multiply $2$ by $2$ .

$- \frac{6}{\sqrt{1 - \frac{9}{x^{4}}} x^{3}}$