Calculus Examples

$\sin^{2} (\ln (x))$

Step 1

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

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\sin (\ln (x))$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sin (\ln (x))]$

Step 1.2

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

$2 u_{1} \frac{d}{d x} [\sin (\ln (x))]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\sin (\ln (x))$ .

$2 \sin (\ln (x)) \frac{d}{d x} [\sin (\ln (x))]$

Step 2

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

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u_{2}$ as $\ln (x)$ .

$2 \sin (\ln (x)) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [\ln (x)])$

Step 2.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$2 \sin (\ln (x)) (\cos (u_{2}) \frac{d}{d x} [\ln (x)])$

Step 2.3

Replace all occurrences of $u_{2}$ with $\ln (x)$ .

$2 \sin (\ln (x)) (\cos (\ln (x)) \frac{d}{d x} [\ln (x)])$

Step 3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$2 \sin (\ln (x)) \cos (\ln (x)) \frac{1}{x}$

Step 4

Combine fractions.

Tap for more steps...

Step 4.1

Combine $\frac{1}{x}$ and $2$ .

$\frac{2}{x} \sin (\ln (x)) \cos (\ln (x))$

Step 4.2

Combine $\frac{2}{x}$ and $\sin (\ln (x))$ .

$\frac{2 \sin (\ln (x))}{x} \cos (\ln (x))$

Step 4.3

Combine $\frac{2 \sin (\ln (x))}{x}$ and $\cos (\ln (x))$ .

$\frac{2 \sin (\ln (x)) \cos (\ln (x))}{x}$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Apply the sine double-angle identity.

$\frac{\sin (2 \ln (x))}{x}$

Step 5.2

Simplify $2 \ln (x)$ by moving $2$ inside the logarithm.

$\frac{\sin (\ln (x^{2}))}{x}$