Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 $g (x) = \tan (x^{3})$ .

Tap for more steps...

Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $\tan (x^{3})$ .

$\frac{d}{d u_{1}} [\arcsin (u_{1})] \frac{d}{d x} [\tan (x^{3})]$

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u_{1}$ with $\tan (x^{3})$ .

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

The derivative of $\tan (u_{2})$ with respect to $u_{2}$ is $\sec^{2} (u_{2})$ .

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

Step 3.2.3

Replace all occurrences of $u_{2}$ with $x^{3}$ .

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

Step 3.3

Differentiate using the Power Rule.

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 3.3.3

Combine fractions.

Tap for more steps...

Step 3.3.3.1

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

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

Step 3.3.3.2

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

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

Step 3.3.3.3

Reorder factors in $3 \sec^{2} (x^{3}) x^{2}$ .

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

Step 4

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

$y' = \frac{3 x^{2} \sec^{2} (x^{3})}{\sqrt{1 - \tan^{2} (x^{3})}}$

Step 5

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

$\frac{d y}{d x} = \frac{3 x^{2} \sec^{2} (x^{3})}{\sqrt{1 - \tan^{2} (x^{3})}}$