Calculus Examples

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplify.

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Apply the distributive property.

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

Reorder the factors of $(2 \cos (x) + 2 \sin (x)) (- \sin (x) + \cos (x))$ .

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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Simplify each term.

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Add parentheses.

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

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

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

Apply the sine double-angle identity.

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

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

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Multiply $2$ by $- 1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

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

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

Apply the sine double-angle identity.

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

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

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

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

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