Calculus Examples

$y = \sin (x) \cos^{2} (x)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sin (x)$ and $g (x) = \cos^{2} (x)$ .

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

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $- 1$ by $2$ .

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

Step 6

Raise $\sin (x)$ to the power of $1$ .

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

Step 7

Raise $\sin (x)$ to the power of $1$ .

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

Step 8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 9

Add $1$ and $1$ .

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

Step 10

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

$- 2 \sin^{2} (x) \cos (x) + \cos^{2} (x) \cos (x)$

Step 11

Multiply $\cos^{2} (x)$ by $\cos (x)$ by adding the exponents.

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Step 11.1

Multiply $\cos^{2} (x)$ by $\cos (x)$ .

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Step 11.1.1

Raise $\cos (x)$ to the power of $1$ .

$- 2 \sin^{2} (x) \cos (x) + \cos^{2} (x) \cos^{1} (x)$

Step 11.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 2 \sin^{2} (x) \cos (x) + {\cos (x)}^{2 + 1}$

Step 11.2

Add $2$ and $1$ .

$- 2 \sin^{2} (x) \cos (x) + \cos^{3} (x)$