Precalculus Examples

$f (x) = x^{3} + x$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

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

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Step 2.1.1

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

$f (x + h) = {(x + h)}^{3} + x + h$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = {(x + h)}^{3} + x + h$

Step 2.1.2.2

Use the Binomial Theorem.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} + x + h$

Step 2.1.2.3

The final answer is $x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} + x + h$ .

$x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} + x + h$

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{2}$ .

$x^{3} + 3 h x^{2} + 3 x h^{2} + h^{3} + x + h$

Step 2.2.2

Move $x$ .

$x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3} + x + h$

Step 2.2.3

Move $x$ .

$x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3} + h + x$

Step 2.2.4

Move $x^{3}$ .

$3 h x^{2} + 3 h^{2} x + h^{3} + x^{3} + h + x$

Step 2.2.5

Move $3 h x^{2}$ .

$3 h^{2} x + h^{3} + 3 h x^{2} + x^{3} + h + x$

Step 2.2.6

Reorder $3 h^{2} x$ and $h^{3}$ .

$h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} + h + x$

Step 2.3

Find the components of the definition.

$f (x + h) = h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} + h + x$

$f (x) = x^{3} + x$

$f (x + h) = h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} + h + x$

$f (x) = x^{3} + x$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} + h + x - (x^{3} + x)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} + h + x - x^{3} - x}{h}$

Step 4.1.2

Subtract $x^{3}$ from $x^{3}$ .

$\frac{h^{3} + 3 h^{2} x + 3 h x^{2} + h + x + 0 - x}{h}$

Step 4.1.3

Add $h^{3}$ and $0$ .

$\frac{h^{3} + 3 h^{2} x + 3 h x^{2} + h + x - x}{h}$

Step 4.1.4

Subtract $x$ from $x$ .

$\frac{h^{3} + 3 h^{2} x + 3 h x^{2} + h + 0}{h}$

Step 4.1.5

Add $h^{3} + 3 h^{2} x + 3 h x^{2} + h$ and $0$ .

$\frac{h^{3} + 3 h^{2} x + 3 h x^{2} + h}{h}$

Step 4.1.6

Factor $h$ out of $h^{3} + 3 h^{2} x + 3 h x^{2} + h$ .

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Step 4.1.6.1

Factor $h$ out of $h^{3}$ .

$\frac{h \cdot h^{2} + 3 h^{2} x + 3 h x^{2} + h}{h}$

Step 4.1.6.2

Factor $h$ out of $3 h^{2} x$ .

$\frac{h (h^{2}) + h (3 h x) + 3 h x^{2} + h}{h}$

Step 4.1.6.3

Factor $h$ out of $3 h x^{2}$ .

$\frac{h (h^{2}) + h (3 h x) + h (3 x^{2}) + h}{h}$

Step 4.1.6.4

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

$\frac{h (h^{2}) + h (3 h x) + h (3 x^{2}) + h^{1}}{h}$

Step 4.1.6.5

Factor $h$ out of $h^{1}$ .

$\frac{h (h^{2}) + h (3 h x) + h (3 x^{2}) + h \cdot 1}{h}$

Step 4.1.6.6

Factor $h$ out of $h (h^{2}) + h (3 h x)$ .

$\frac{h (h^{2} + 3 h x) + h (3 x^{2}) + h \cdot 1}{h}$

Step 4.1.6.7

Factor $h$ out of $h (h^{2} + 3 h x) + h (3 x^{2})$ .

$\frac{h (h^{2} + 3 h x + 3 x^{2}) + h \cdot 1}{h}$

Step 4.1.6.8

Factor $h$ out of $h (h^{2} + 3 h x + 3 x^{2}) + h \cdot 1$ .

$\frac{h (h^{2} + 3 h x + 3 x^{2} + 1)}{h}$

Step 4.2

Reduce the expression by cancelling the common factors.

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Step 4.2.1

Cancel the common factor of $h$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{h (h^{2} + 3 h x + 3 x^{2} + 1)}{h}$

Step 4.2.1.2

Divide $h^{2} + 3 h x + 3 x^{2} + 1$ by $1$ .

$h^{2} + 3 h x + 3 x^{2} + 1$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Move $h$ .

$h^{2} + 3 x h + 3 x^{2} + 1$

Step 4.2.2.2

Move $h^{2}$ .

$3 x h + 3 x^{2} + h^{2} + 1$

Step 4.2.2.3

Reorder $3 x h$ and $3 x^{2}$ .

$3 x^{2} + 3 x h + h^{2} + 1$

Step 5