Calculus Examples

$4 \sin (x) \cos (x)$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 \sin (x) \cos (x)$ with respect to $x$ is $4 \frac{d}{d x} [\sin (x) \cos (x)]$ .

$4 \frac{d}{d x} [\sin (x) \cos (x)]$

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Add $1$ and $1$ .

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

Step 13

Simplify.

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Step 13.1

Apply the distributive property.

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

Step 13.2

Multiply $- 1$ by $4$ .

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

Step 13.3

Rewrite $4 \cos^{2} (x)$ as ${(2 \cos (x))}^{2}$ .

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

Step 13.4

Rewrite $4 \sin^{2} (x)$ as ${(2 \sin (x))}^{2}$ .

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

Step 13.5

Reorder $- {(2 \sin (x))}^{2}$ and ${(2 \cos (x))}^{2}$ .

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

Step 13.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 \cos (x)$ and $b = 2 \sin (x)$ .

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

Step 13.7

Multiply $2$ by $- 1$ .

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

Step 13.8

Expand $(2 \cos (x) + 2 \sin (x)) (2 \cos (x) - 2 \sin (x))$ using the FOIL Method.

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Step 13.8.1

Apply the distributive property.

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

Step 13.8.2

Apply the distributive property.

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

Step 13.8.3

Apply the distributive property.

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

Step 13.9

Combine the opposite terms in $2 \cos (x) (2 \cos (x)) + 2 \cos (x) (- 2 \sin (x)) + 2 \sin (x) (2 \cos (x)) + 2 \sin (x) (- 2 \sin (x))$ .

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Step 13.9.1

Reorder the factors in the terms $2 \cos (x) (- 2 \sin (x))$ and $2 \sin (x) (2 \cos (x))$ .

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

Step 13.9.2

Add $- 2 \cdot 2 \cos (x) \sin (x)$ and $2 \cdot 2 \cos (x) \sin (x)$ .

$2 \cos (x) (2 \cos (x)) + 0 + 2 \sin (x) (- 2 \sin (x))$

Step 13.9.3

Add $2 \cos (x) (2 \cos (x))$ and $0$ .

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

Step 13.10

Simplify each term.

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Step 13.10.1

Multiply $2 \cos (x) (2 \cos (x))$ .

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Step 13.10.1.1

Multiply $2$ by $2$ .

$4 \cos (x) \cos (x) + 2 \sin (x) (- 2 \sin (x))$

Step 13.10.1.2

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

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

Step 13.10.1.3

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

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

Step 13.10.1.4

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

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

Step 13.10.1.5

Add $1$ and $1$ .

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

Step 13.10.2

Multiply $2 \sin (x) (- 2 \sin (x))$ .

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Step 13.10.2.1

Multiply $- 2$ by $2$ .

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

Step 13.10.2.2

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

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

Step 13.10.2.3

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

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

Step 13.10.2.4

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

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

Step 13.10.2.5

Add $1$ and $1$ .

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