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 $f (A) = \sin (A)$ and $g (A) = 2 A$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $2 A$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $2 A$ .

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

Step 2

Differentiate.

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Step 2.1

Since $2$ is constant with respect to $A$ , the derivative of $2 A$ with respect to $A$ is $2 \frac{d}{d A} [A]$ .

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

Step 2.2

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

$\cos (2 A) (2 \cdot 1)$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $2$ by $1$ .

$\cos (2 A) \cdot 2$

Step 2.3.2

Move $2$ to the left of $\cos (2 A)$ .

$2 \cos (2 A)$