Calculus Examples

$\frac{\sin (t)}{1 - \cos (t)}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = \sin (t)$ and $g (t) = 1 - \cos (t)$ .

$\frac{(1 - \cos (t)) \frac{d}{d t} [\sin (t)] - \sin (t) \frac{d}{d t} [1 - \cos (t)]}{{(1 - \cos (t))}^{2}}$

Step 2

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

$\frac{(1 - \cos (t)) \cos (t) - \sin (t) \frac{d}{d t} [1 - \cos (t)]}{{(1 - \cos (t))}^{2}}$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $1 - \cos (t)$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [- \cos (t)]$ .

$\frac{(1 - \cos (t)) \cos (t) - \sin (t) (\frac{d}{d t} [1] + \frac{d}{d t} [- \cos (t)])}{{(1 - \cos (t))}^{2}}$

Step 3.2

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$\frac{(1 - \cos (t)) \cos (t) - \sin (t) (0 + \frac{d}{d t} [- \cos (t)])}{{(1 - \cos (t))}^{2}}$

Step 3.3

Add $0$ and $\frac{d}{d t} [- \cos (t)]$ .

$\frac{(1 - \cos (t)) \cos (t) - \sin (t) \frac{d}{d t} [- \cos (t)]}{{(1 - \cos (t))}^{2}}$

Step 3.4

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

$\frac{(1 - \cos (t)) \cos (t) - \sin (t) (- \frac{d}{d t} [\cos (t)])}{{(1 - \cos (t))}^{2}}$

Step 3.5

Multiply.

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Step 3.5.1

Multiply $- 1$ by $- 1$ .

$\frac{(1 - \cos (t)) \cos (t) + 1 \sin (t) \frac{d}{d t} [\cos (t)]}{{(1 - \cos (t))}^{2}}$

Step 3.5.2

Multiply $\sin (t)$ by $1$ .

$\frac{(1 - \cos (t)) \cos (t) + \sin (t) \frac{d}{d t} [\cos (t)]}{{(1 - \cos (t))}^{2}}$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Add $1$ and $1$ .

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Simplify the numerator.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Multiply $\cos (t)$ by $1$ .

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

Step 9.2.1.2

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

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Step 9.2.1.2.1

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

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

Step 9.2.1.2.2

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

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

Step 9.2.1.2.3

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

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

Step 9.2.1.2.4

Add $1$ and $1$ .

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

Step 9.2.2

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

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

Step 9.2.3

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

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

Step 9.2.4

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

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

Step 9.2.5

Rearrange terms.

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

Step 9.2.6

Apply pythagorean identity.

$\frac{\cos (t) - 1 \cdot 1}{{(1 - \cos (t))}^{2}}$

Step 9.2.7

Multiply $- 1$ by $1$ .

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

Step 9.3

Combine terms.

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Step 9.3.1

Cancel the common factor of $\cos (t) - 1$ and ${(1 - \cos (t))}^{2}$ .

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Step 9.3.1.1

Factor $- 1$ out of $\cos (t)$ .

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

Step 9.3.1.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 9.3.1.3

Factor $- 1$ out of $- 1 (- \cos (t)) - 1 (1)$ .

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

Step 9.3.1.4

Reorder terms.

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

Step 9.3.1.5

Factor $1 - \cos (t)$ out of $- 1 (1 - \cos (t))$ .

$\frac{(1 - \cos (t)) \cdot - 1}{{(1 - \cos (t))}^{2}}$

Step 9.3.1.6

Cancel the common factors.

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Step 9.3.1.6.1

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

$\frac{(1 - \cos (t)) \cdot - 1}{(1 - \cos (t)) (1 - \cos (t))}$

Step 9.3.1.6.2

Cancel the common factor.

$\frac{(1 - \cos (t)) \cdot - 1}{(1 - \cos (t)) (1 - \cos (t))}$

Step 9.3.1.6.3

Rewrite the expression.

$\frac{- 1}{1 - \cos (t)}$

Step 9.3.2

Move the negative in front of the fraction.

$- \frac{1}{1 - \cos (t)}$