Calculus Examples

$y = 3 (\cos (x) + \sin (x)) \cdot (\cos (x) - \sin (x))$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 (\cos (x) + \sin (x)) (\cos (x) - \sin (x))$ with respect to $x$ is $3 \frac{d}{d x} [(\cos (x) + \sin (x)) (\cos (x) - \sin (x))]$ .

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

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 $f (x) = \cos (x) + \sin (x)$ and $g (x) = \cos (x) - \sin (x)$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

$3 ((\cos (x) + \sin (x)) (- \sin (x) - \cos (x)) + (\cos (x) - \sin (x)) (- \sin (x) + \cos (x)))$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$3 ((\cos (x) + \sin (x)) (- \sin (x) - \cos (x))) + 3 ((\cos (x) - \sin (x)) (- \sin (x) + \cos (x)))$

Step 10.2

Remove parentheses.

$3 (\cos (x) + \sin (x)) (- \sin (x) - \cos (x)) + 3 (\cos (x) - \sin (x)) (- \sin (x) + \cos (x))$

Step 10.3

Reorder terms.

$3 (- \sin (x) - \cos (x)) (\cos (x) + \sin (x)) + 3 (- \sin (x) + \cos (x)) (\cos (x) - \sin (x))$

Step 10.4

Simplify each term.

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Step 10.4.1

Apply the distributive property.

$(3 (- \sin (x)) + 3 (- \cos (x))) (\cos (x) + \sin (x)) + 3 (- \sin (x) + \cos (x)) (\cos (x) - \sin (x))$

Step 10.4.2

Multiply $- 1$ by $3$ .

$(- 3 \sin (x) + 3 (- \cos (x))) (\cos (x) + \sin (x)) + 3 (- \sin (x) + \cos (x)) (\cos (x) - \sin (x))$

Step 10.4.3

Multiply $- 1$ by $3$ .

$(- 3 \sin (x) - 3 \cos (x)) (\cos (x) + \sin (x)) + 3 (- \sin (x) + \cos (x)) (\cos (x) - \sin (x))$

Step 10.4.4

Expand $(- 3 \sin (x) - 3 \cos (x)) (\cos (x) + \sin (x))$ using the FOIL Method.

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Step 10.4.4.1

Apply the distributive property.

$- 3 \sin (x) (\cos (x) + \sin (x)) - 3 \cos (x) (\cos (x) + \sin (x)) + 3 (- \sin (x) + \cos (x)) (\cos (x) - \sin (x))$

Step 10.4.4.2

Apply the distributive property.

$- 3 \sin (x) \cos (x) - 3 \sin (x) \sin (x) - 3 \cos (x) (\cos (x) + \sin (x)) + 3 (- \sin (x) + \cos (x)) (\cos (x) - \sin (x))$

Step 10.4.4.3

Apply the distributive property.

$- 3 \sin (x) \cos (x) - 3 \sin (x) \sin (x) - 3 \cos (x) \cos (x) - 3 \cos (x) \sin (x) + 3 (- \sin (x) + \cos (x)) (\cos (x) - \sin (x))$

Step 10.4.5

Simplify and combine like terms.

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Step 10.4.5.1

Simplify each term.

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Step 10.4.5.1.1

Multiply $- 3 \sin (x) \sin (x)$ .

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Step 10.4.5.1.1.1

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

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

Step 10.4.5.1.1.2

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

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

Step 10.4.5.1.1.3

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

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

Step 10.4.5.1.1.4

Add $1$ and $1$ .

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

Step 10.4.5.1.2

Multiply $- 3 \cos (x) \cos (x)$ .

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Step 10.4.5.1.2.1

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

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

Step 10.4.5.1.2.2

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

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

Step 10.4.5.1.2.3

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

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

Step 10.4.5.1.2.4

Add $1$ and $1$ .

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

Step 10.4.5.2

Reorder the factors of $- 3 \sin (x) \cos (x)$ .

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

Step 10.4.5.3

Subtract $3 \cos (x) \sin (x)$ from $- 3 \cos (x) \sin (x)$ .

$- 6 \cos (x) \sin (x) - 3 \sin^{2} (x) - 3 \cos^{2} (x) + 3 (- \sin (x) + \cos (x)) (\cos (x) - \sin (x))$

Step 10.4.6

Factor $- 3$ out of $- 3 \sin^{2} (x)$ .

$- 6 \cos (x) \sin (x) - 3 (\sin^{2} (x)) - 3 \cos^{2} (x) + 3 (- \sin (x) + \cos (x)) (\cos (x) - \sin (x))$

Step 10.4.7

Factor $- 3$ out of $- 3 \cos^{2} (x)$ .

$- 6 \cos (x) \sin (x) - 3 (\sin^{2} (x)) - 3 (\cos^{2} (x)) + 3 (- \sin (x) + \cos (x)) (\cos (x) - \sin (x))$

Step 10.4.8

Factor $- 3$ out of $- 3 (\sin^{2} (x)) - 3 (\cos^{2} (x))$ .

$- 6 \cos (x) \sin (x) - 3 (\sin^{2} (x) + \cos^{2} (x)) + 3 (- \sin (x) + \cos (x)) (\cos (x) - \sin (x))$

Step 10.4.9

Apply pythagorean identity.

$- 6 \cos (x) \sin (x) - 3 \cdot 1 + 3 (- \sin (x) + \cos (x)) (\cos (x) - \sin (x))$

Step 10.4.10

Multiply $- 3$ by $1$ .

$- 6 \cos (x) \sin (x) - 3 + 3 (- \sin (x) + \cos (x)) (\cos (x) - \sin (x))$

Step 10.4.11

Apply the distributive property.

$- 6 \cos (x) \sin (x) - 3 + (3 (- \sin (x)) + 3 \cos (x)) (\cos (x) - \sin (x))$

Step 10.4.12

Multiply $- 1$ by $3$ .

$- 6 \cos (x) \sin (x) - 3 + (- 3 \sin (x) + 3 \cos (x)) (\cos (x) - \sin (x))$

Step 10.4.13

Expand $(- 3 \sin (x) + 3 \cos (x)) (\cos (x) - \sin (x))$ using the FOIL Method.

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Step 10.4.13.1

Apply the distributive property.

$- 6 \cos (x) \sin (x) - 3 - 3 \sin (x) (\cos (x) - \sin (x)) + 3 \cos (x) (\cos (x) - \sin (x))$

Step 10.4.13.2

Apply the distributive property.

$- 6 \cos (x) \sin (x) - 3 - 3 \sin (x) \cos (x) - 3 \sin (x) (- \sin (x)) + 3 \cos (x) (\cos (x) - \sin (x))$

Step 10.4.13.3

Apply the distributive property.

$- 6 \cos (x) \sin (x) - 3 - 3 \sin (x) \cos (x) - 3 \sin (x) (- \sin (x)) + 3 \cos (x) \cos (x) + 3 \cos (x) (- \sin (x))$

Step 10.4.14

Simplify and combine like terms.

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Step 10.4.14.1

Simplify each term.

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Step 10.4.14.1.1

Multiply $- 3 \sin (x) (- \sin (x))$ .

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Step 10.4.14.1.1.1

Multiply $- 1$ by $- 3$ .

$- 6 \cos (x) \sin (x) - 3 - 3 \sin (x) \cos (x) + 3 \sin (x) \sin (x) + 3 \cos (x) \cos (x) + 3 \cos (x) (- \sin (x))$

Step 10.4.14.1.1.2

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

$- 6 \cos (x) \sin (x) - 3 - 3 \sin (x) \cos (x) + 3 (\sin^{1} (x) \sin (x)) + 3 \cos (x) \cos (x) + 3 \cos (x) (- \sin (x))$

Step 10.4.14.1.1.3

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

$- 6 \cos (x) \sin (x) - 3 - 3 \sin (x) \cos (x) + 3 (\sin^{1} (x) \sin^{1} (x)) + 3 \cos (x) \cos (x) + 3 \cos (x) (- \sin (x))$

Step 10.4.14.1.1.4

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

$- 6 \cos (x) \sin (x) - 3 - 3 \sin (x) \cos (x) + 3 {\sin (x)}^{1 + 1} + 3 \cos (x) \cos (x) + 3 \cos (x) (- \sin (x))$

Step 10.4.14.1.1.5

Add $1$ and $1$ .

$- 6 \cos (x) \sin (x) - 3 - 3 \sin (x) \cos (x) + 3 \sin^{2} (x) + 3 \cos (x) \cos (x) + 3 \cos (x) (- \sin (x))$

Step 10.4.14.1.2

Multiply $3 \cos (x) \cos (x)$ .

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Step 10.4.14.1.2.1

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

$- 6 \cos (x) \sin (x) - 3 - 3 \sin (x) \cos (x) + 3 \sin^{2} (x) + 3 (\cos^{1} (x) \cos (x)) + 3 \cos (x) (- \sin (x))$

Step 10.4.14.1.2.2

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

$- 6 \cos (x) \sin (x) - 3 - 3 \sin (x) \cos (x) + 3 \sin^{2} (x) + 3 (\cos^{1} (x) \cos^{1} (x)) + 3 \cos (x) (- \sin (x))$

Step 10.4.14.1.2.3

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

$- 6 \cos (x) \sin (x) - 3 - 3 \sin (x) \cos (x) + 3 \sin^{2} (x) + 3 {\cos (x)}^{1 + 1} + 3 \cos (x) (- \sin (x))$

Step 10.4.14.1.2.4

Add $1$ and $1$ .

$- 6 \cos (x) \sin (x) - 3 - 3 \sin (x) \cos (x) + 3 \sin^{2} (x) + 3 \cos^{2} (x) + 3 \cos (x) (- \sin (x))$

Step 10.4.14.1.3

Multiply $- 1$ by $3$ .

$- 6 \cos (x) \sin (x) - 3 - 3 \sin (x) \cos (x) + 3 \sin^{2} (x) + 3 \cos^{2} (x) - 3 \cos (x) \sin (x)$

Step 10.4.14.2

Reorder the factors of $- 3 \sin (x) \cos (x)$ .

$- 6 \cos (x) \sin (x) - 3 - 3 \cos (x) \sin (x) + 3 \sin^{2} (x) + 3 \cos^{2} (x) - 3 \cos (x) \sin (x)$

Step 10.4.14.3

Subtract $3 \cos (x) \sin (x)$ from $- 3 \cos (x) \sin (x)$ .

$- 6 \cos (x) \sin (x) - 3 - 6 \cos (x) \sin (x) + 3 \sin^{2} (x) + 3 \cos^{2} (x)$

Step 10.4.15

Factor $3$ out of $3 \sin^{2} (x)$ .

$- 6 \cos (x) \sin (x) - 3 - 6 \cos (x) \sin (x) + 3 (\sin^{2} (x)) + 3 \cos^{2} (x)$

Step 10.4.16

Factor $3$ out of $3 \cos^{2} (x)$ .

$- 6 \cos (x) \sin (x) - 3 - 6 \cos (x) \sin (x) + 3 (\sin^{2} (x)) + 3 (\cos^{2} (x))$

Step 10.4.17

Factor $3$ out of $3 (\sin^{2} (x)) + 3 (\cos^{2} (x))$ .

$- 6 \cos (x) \sin (x) - 3 - 6 \cos (x) \sin (x) + 3 (\sin^{2} (x) + \cos^{2} (x))$

Step 10.4.18

Apply pythagorean identity.

$- 6 \cos (x) \sin (x) - 3 - 6 \cos (x) \sin (x) + 3 \cdot 1$

Step 10.4.19

Multiply $3$ by $1$ .

$- 6 \cos (x) \sin (x) - 3 - 6 \cos (x) \sin (x) + 3$

Step 10.5

Combine the opposite terms in $- 6 \cos (x) \sin (x) - 3 - 6 \cos (x) \sin (x) + 3$ .

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Step 10.5.1

Add $- 3$ and $3$ .

$- 6 \cos (x) \sin (x) - 6 \cos (x) \sin (x) + 0$

Step 10.5.2

Add $- 6 \cos (x) \sin (x) - 6 \cos (x) \sin (x)$ and $0$ .

$- 6 \cos (x) \sin (x) - 6 \cos (x) \sin (x)$

Step 10.6

Subtract $6 \cos (x) \sin (x)$ from $- 6 \cos (x) \sin (x)$ .

$- 12 \cos (x) \sin (x)$