Calculus Examples

$\frac{1}{2} \sin^{2} (x)$

Step 1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2} \sin^{2} (x)$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\sin^{2} (x)]$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Move $2$ to the left of $\sin (x)$ .

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

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Divide $\sin (x)$ by $1$ .

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

Step 4

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

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

Step 5

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

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