Precalculus Examples

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

Step 1

Write $\frac{f (x + h) - f (x)}{h}$ as a function.

$f (x) = \frac{f (x + h) - f (x)}{h}$

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

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

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Step 3.1.1

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

$f (x + h) = \frac{f ((x + h) + h) - f (x + h)}{h}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify the numerator.

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Step 3.1.2.1.1

Factor $f$ out of $f (x + h + h) - f (x + h)$ .

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Step 3.1.2.1.1.1

Factor $f$ out of $- f (x + h)$ .

$f (x + h) = \frac{f (x + h + h) + f (- 1 (x + h))}{h}$

Step 3.1.2.1.1.2

Factor $f$ out of $f (x + h + h) + f (- 1 (x + h))$ .

$f (x + h) = \frac{f (x + h + h - 1 (x + h))}{h}$

Step 3.1.2.1.2

Apply the distributive property.

$f (x + h) = \frac{f (x + h + h - 1 x - 1 h)}{h}$

Step 3.1.2.1.3

Rewrite $- 1 x$ as $- x$ .

$f (x + h) = \frac{f (x + h + h - x - 1 h)}{h}$

Step 3.1.2.1.4

Rewrite $- 1 h$ as $- h$ .

$f (x + h) = \frac{f (x + h + h - x - h)}{h}$

Step 3.1.2.1.5

Subtract $x$ from $x$ .

$f (x + h) = \frac{f (h + h + 0 - h)}{h}$

Step 3.1.2.1.6

Add $h$ and $h$ .

$f (x + h) = \frac{f (2 h + 0 - h)}{h}$

Step 3.1.2.1.7

Add $2 h$ and $0$ .

$f (x + h) = \frac{f (2 h - h)}{h}$

Step 3.1.2.1.8

Subtract $h$ from $2 h$ .

$f (x + h) = \frac{f h}{h}$

Step 3.1.2.2

Cancel the common factor of $h$ .

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Step 3.1.2.2.1

Cancel the common factor.

$f (x + h) = \frac{f h}{h}$

Step 3.1.2.2.2

Divide $f$ by $1$ .

$f (x + h) = f$

Step 3.1.2.3

The final answer is $f$ .

$f$

Step 3.2

Find the components of the definition.

$f (x + h) = f$

$f (x) = \frac{f x + f h - f (x)}{h}$

$f (x + h) = f$

$f (x) = \frac{f x + f h - f (x)}{h}$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{f - (\frac{f x + f h - f (x)}{h})}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

To write $f$ as a fraction with a common denominator, multiply by $\frac{h}{h}$ .

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

Step 5.1.2

Combine the numerators over the common denominator.

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

Step 5.1.3

Rewrite $\frac{f h - (f x + f h - f (x))}{h}$ in a factored form.

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Step 5.1.3.1

Apply the distributive property.

$\frac{\frac{f h - (f x) - (f h) - - f (x)}{h}}{h}$

Step 5.1.3.2

Multiply $- - f (x)$ .

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Step 5.1.3.2.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{f h - f x - f h + 1 f (x)}{h}}{h}$

Step 5.1.3.2.2

Multiply $f (x)$ by $1$ .

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

Step 5.1.3.3

Subtract $f h$ from $f h$ .

$\frac{\frac{- f x + 0 + f (x)}{h}}{h}$

Step 5.1.3.4

Add $- f x$ and $0$ .

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

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{- f x + f (x)}{h} \cdot \frac{1}{h}$

Step 5.3

Multiply $\frac{- f x + f (x)}{h} \cdot \frac{1}{h}$ .

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Step 5.3.1

Multiply $\frac{- f x + f (x)}{h}$ by $\frac{1}{h}$ .

$\frac{- f x + f (x)}{h \cdot h}$

Step 5.3.2

Raise $h$ to the power of $1$ .

$\frac{- f x + f (x)}{h^{1} h}$

Step 5.3.3

Raise $h$ to the power of $1$ .

$\frac{- f x + f (x)}{h^{1} h^{1}}$

Step 5.3.4

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

$\frac{- f x + f (x)}{h^{1 + 1}}$

Step 5.3.5

Add $1$ and $1$ .

$\frac{- f x + f (x)}{h^{2}}$

Step 5.4

Factor $- 1$ out of $- f x$ .

$\frac{- (f x) + f (x)}{h^{2}}$

Step 5.5

Factor $- 1$ out of $f (x)$ .

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

Step 5.6

Factor $- 1$ out of $- (f x) - 1 (- f (x))$ .

$\frac{- (f x - f (x))}{h^{2}}$

Step 5.7

Simplify the expression.

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Step 5.7.1

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

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

Step 5.7.2

Move the negative in front of the fraction.

$- \frac{f x - f (x)}{h^{2}}$

Step 6