Calculus Examples

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

Step 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 $g (x) = \sin (x) + \cos (x)$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

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

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Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Apply the sine double-angle identity.

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

Step 5.3.1.2

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

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Step 5.3.1.2.1

Multiply $- 1$ by $2$ .

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

Step 5.3.1.2.2

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

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

Step 5.3.1.2.3

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

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

Step 5.3.1.2.4

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

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

Step 5.3.1.2.5

Add $1$ and $1$ .

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

Step 5.3.1.3

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

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Step 5.3.1.3.1

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

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

Step 5.3.1.3.2

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

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

Step 5.3.1.3.3

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

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

Step 5.3.1.3.4

Add $1$ and $1$ .

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

Step 5.3.1.4

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

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

Step 5.3.1.5

Add parentheses.

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

Step 5.3.1.6

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

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

Step 5.3.1.7

Apply the sine double-angle identity.

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

Step 5.3.2

Subtract $\sin (2 x)$ from $\sin (2 x)$ .

$0 - 2 \sin^{2} (x) + 2 \cos^{2} (x)$

Step 5.3.3

Subtract $2 \sin^{2} (x)$ from $0$ .

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