Calculus Examples

$\cos^{2} (A)$

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 $g (A) = \cos (A)$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Multiply $- 1$ by $2$ .

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