Calculus Examples

$y = e^{\ln (\sin (x))} \sin (x)$

Step 1

Exponentiation and log are inverse functions.

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

Step 2

Raise $\sin (x)$ to the power of $1$ .

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

Step 3

Raise $\sin (x)$ to the power of $1$ .

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

Step 4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [{\sin (x)}^{1 + 1}]$

Step 5

Add $1$ and $1$ .

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

Step 6

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 $g (x) = \sin (x)$ .

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Step 6.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

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

Step 6.2

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

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

Step 6.3

Replace all occurrences of $u$ with $\sin (x)$ .

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

Step 7

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$2 \sin (x) \cos (x)$

Step 8

Simplify.

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Step 8.1

Reorder the factors of $2 \sin (x) \cos (x)$ .

$2 \cos (x) \sin (x)$

Step 8.2

Reorder $2 \cos (x)$ and $\sin (x)$ .

$\sin (x) (2 \cos (x))$

Step 8.3

Reorder $\sin (x)$ and $2$ .

$2 \cdot \sin (x) \cos (x)$

Step 8.4

Apply the sine double-angle identity.

$\sin (2 x)$