Calculus Examples

$\sin^{2} (A)$

Step 1

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

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

$2 \sin (A) \cos (A)$

Step 3

Simplify.

Tap for more steps...

Step 3.1

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

$2 \cos (A) \sin (A)$

Step 3.2

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

$\sin (A) (2 \cos (A))$

Step 3.3

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

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

Step 3.4

Apply the sine double-angle identity.

$\sin (2 A)$