Calculus Examples

$\sin^{3} (x) \cos (x)$

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 $f (x) = \sin^{3} (x)$ and $g (x) = \cos (x)$ .

$\sin^{3} (x) \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [\sin^{3} (x)]$

Step 2

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

$\sin^{3} (x) (- \sin (x)) + \cos (x) \frac{d}{d x} [\sin^{3} (x)]$

Step 3

Multiply $\sin^{3} (x)$ by $\sin (x)$ by adding the exponents.

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Step 3.1

Move $\sin (x)$ .

$\sin (x) \sin^{3} (x) \cdot - 1 + \cos (x) \frac{d}{d x} [\sin^{3} (x)]$

Step 3.2

Multiply $\sin (x)$ by $\sin^{3} (x)$ .

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Step 3.2.1

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

$\sin^{1} (x) \sin^{3} (x) \cdot - 1 + \cos (x) \frac{d}{d x} [\sin^{3} (x)]$

Step 3.2.2

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

${\sin (x)}^{1 + 3} \cdot - 1 + \cos (x) \frac{d}{d x} [\sin^{3} (x)]$

Step 3.3

Add $1$ and $3$ .

$\sin^{4} (x) \cdot - 1 + \cos (x) \frac{d}{d x} [\sin^{3} (x)]$

Step 4

Simplify the expression.

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Step 4.1

Move $- 1$ to the left of $\sin^{4} (x)$ .

$- 1 \cdot \sin^{4} (x) + \cos (x) \frac{d}{d x} [\sin^{3} (x)]$

Step 4.2

Rewrite $- 1 \sin^{4} (x)$ as $- \sin^{4} (x)$ .

$- \sin^{4} (x) + \cos (x) \frac{d}{d x} [\sin^{3} (x)]$

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \sin (x)$ .

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Step 5.1

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

$- \sin^{4} (x) + \cos (x) (\frac{d}{d u} [u^{3}] \frac{d}{d x} [\sin (x)])$

Step 5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$- \sin^{4} (x) + \cos (x) (3 u^{2} \frac{d}{d x} [\sin (x)])$

Step 5.3

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

$- \sin^{4} (x) + \cos (x) (3 \sin^{2} (x) \frac{d}{d x} [\sin (x)])$

Step 6

Move $3$ to the left of $\cos (x)$ .

$- \sin^{4} (x) + 3 \cdot \cos (x) \sin^{2} (x) \frac{d}{d x} [\sin (x)]$

Step 7

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

$- \sin^{4} (x) + 3 \cos (x) \sin^{2} (x) \cos (x)$

Step 8

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

$- \sin^{4} (x) + 3 (\cos^{1} (x) \cos (x)) \sin^{2} (x)$

Step 9

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

$- \sin^{4} (x) + 3 (\cos^{1} (x) \cos^{1} (x)) \sin^{2} (x)$

Step 10

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

$- \sin^{4} (x) + 3 {\cos (x)}^{1 + 1} \sin^{2} (x)$

Step 11

Add $1$ and $1$ .

$- \sin^{4} (x) + 3 \cos^{2} (x) \sin^{2} (x)$

Step 12

Reorder terms.

$3 \cos^{2} (x) \sin^{2} (x) - \sin^{4} (x)$