Calculus Examples

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

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \cos (x)$ and $g (x) = \sin (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.1.1.2

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

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Step 4.1.1.2.1

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

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

Step 4.1.1.2.2

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

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

Step 4.1.1.2.3

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

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

Step 4.1.1.2.4

Add $1$ and $1$ .

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

Step 4.1.1.3

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

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Step 4.1.1.3.1

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

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

Step 4.1.1.3.2

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

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

Step 4.1.1.3.3

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

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

Step 4.1.1.3.4

Add $1$ and $1$ .

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

Step 4.1.2

Factor $- 1$ out of $- \sin^{2} (x)$ .

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

Step 4.1.3

Factor $- 1$ out of $- \cos^{2} (x)$ .

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

Step 4.1.4

Factor $- 1$ out of $- (\sin^{2} (x)) - (\cos^{2} (x))$ .

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

Step 4.1.5

Apply pythagorean identity.

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

Step 4.1.6

Multiply $- 1$ by $1$ .

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

Step 4.2

Combine terms.

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Step 4.2.1

Move the negative in front of the fraction.

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

Step 4.2.2

Convert from $\frac{1}{\sin^{2} (x)}$ to $\csc^{2} (x)$ .

$- \csc^{2} (x)$