Precalculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Use the Binomial Theorem.

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

Step 2.1.2.1.2

Apply the distributive property.

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

Step 2.1.2.2

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

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

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{2}$ .

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

Step 2.2.2

Move $x$ .

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

Step 2.2.3

Move $3 x$ .

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

Step 2.2.4

Move $x^{3}$ .

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

Step 2.2.5

Move $3 h x^{2}$ .

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

Step 2.2.6

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

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

Step 2.3

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

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.

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

Step 4.1.2

Multiply $3$ by $- 1$ .

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

Subtract $3 x$ from $3 x$ .

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

Step 4.1.6

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

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

Step 4.1.7

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

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Step 4.1.7.1

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

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

Step 4.1.7.2

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

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

Step 4.1.7.3

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

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

Step 4.1.7.4

Factor $h$ out of $3 h$ .

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

Step 4.1.7.5

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

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

Step 4.1.7.6

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

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

Step 4.1.7.7

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

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

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.

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

Step 4.2.1.2

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

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

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Move $h$ .

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

Step 4.2.2.2

Move $h^{2}$ .

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

Step 4.2.2.3

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

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

Step 5