Calculus Examples

$f (x) = \sin^{2} (x) + \sin (x)$

Step 1

By the Sum Rule, the derivative of $\sin^{2} (x) + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [\sin^{2} (x)] + \frac{d}{d x} [\sin (x)]$ .

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

Step 2

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

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Step 2.1

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 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Simplify each term.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

Apply the sine double-angle identity.

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