Calculus Examples

$\frac{\cos (x)}{\csc (x)}$

Step 1

Rewrite $\csc (x)$ in terms of sines and cosines.

$\frac{d}{d x} [\frac{\cos (x)}{\frac{1}{\sin (x)}}]$

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Add $1$ and $1$ .

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

Add $1$ and $1$ .

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

Step 14

Simplify.

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Step 14.1

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

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

Step 14.2

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

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Step 14.2.1

Apply the distributive property.

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

Step 14.2.2

Apply the distributive property.

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

Step 14.2.3

Apply the distributive property.

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

Step 14.3

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

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Step 14.3.1

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

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

Step 14.3.2

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

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

Step 14.3.3

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

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

Step 14.4

Simplify each term.

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Step 14.4.1

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

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Step 14.4.1.1

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

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

Step 14.4.1.2

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

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

Step 14.4.1.3

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

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

Step 14.4.1.4

Add $1$ and $1$ .

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

Step 14.4.2

Rewrite using the commutative property of multiplication.

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

Step 14.4.3

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

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Step 14.4.3.1

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

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

Step 14.4.3.2

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

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

Step 14.4.3.3

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

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

Step 14.4.3.4

Add $1$ and $1$ .

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

Step 14.5

Apply the cosine double-angle identity.

$\cos (2 x)$