Precalculus Examples

$h (t) = \cot (t)$

Step 1

Consider the difference quotient formula.

$\frac{f (t + h) - f t}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = t + h$ .

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Step 2.1.1

Replace the variable $t$ with $t + h$ in the expression.

$h (t + h) = \cot (t + h)$

Step 2.1.2

The final answer is $\cot (t + h)$ .

$\cot (t + h)$

Step 2.2

Find the components of the definition.

$h (t + h) = \cot (t + h)$

$h (t) = \cot (t)$

$h (t + h) = \cot (t + h)$

$h (t) = \cot (t)$

Step 3

Plug in the components.

$\frac{h (t + h) - h t}{h} = \frac{\cot (t + h) - (\cot (t))}{h}$

Step 4

Simplify.

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Step 4.1

$\frac{\frac{\cos (t + h)}{\sin (t + h)} - (\frac{\cos (t)}{\sin (t)})}{h}$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the numerator.

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Step 4.2.1.1

Convert from $\frac{\cos (t)}{\sin (t)}$ to $\cot (t)$ .

$\frac{\frac{\cos (t + h)}{\sin (t + h)} - \cot (t)}{h}$

Step 4.2.1.2

To write $- \cot (t)$ as a fraction with a common denominator, multiply by $\frac{\sin (t + h)}{\sin (t + h)}$ .

$\frac{\frac{\cos (t + h)}{\sin (t + h)} - \cot (t) \cdot \frac{\sin (t + h)}{\sin (t + h)}}{h}$

Step 4.2.1.3

Combine $- \cot (t)$ and $\frac{\sin (t + h)}{\sin (t + h)}$ .

$\frac{\frac{\cos (t + h)}{\sin (t + h)} + \frac{- \cot (t) \sin (t + h)}{\sin (t + h)}}{h}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$\frac{\frac{\cos (t + h) - \cot (t) \sin (t + h)}{\sin (t + h)}}{h}$

Step 4.2.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\cos (t + h) - \cot (t) \sin (t + h)}{\sin (t + h)} \cdot \frac{1}{h}$

Step 4.2.3

Multiply $\frac{\cos (t + h) - \cot (t) \sin (t + h)}{\sin (t + h)}$ by $\frac{1}{h}$ .

$\frac{\cos (t + h) - \cot (t) \sin (t + h)}{\sin (t + h) h}$

Step 4.2.4

Reorder factors in $\frac{\cos (t + h) - \cot (t) \sin (t + h)}{\sin (t + h) h}$ .

$\frac{\cos (t + h) - \cot (t) \sin (t + h)}{h \sin (t + h)}$

Step 5