Precalculus Examples

$y = x^{3}$

Step 1

Write $y = x^{3}$ as a function.

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

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.

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

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Use the Binomial Theorem.

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

Step 3.1.2.2

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

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

Step 3.2

Reorder.

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Step 3.2.1

Move $x^{2}$ .

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

Step 3.2.2

Move $x$ .

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

Step 3.2.3

Move $x^{3}$ .

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

Step 3.2.4

Move $3 h x^{2}$ .

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

Step 3.2.5

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

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

Step 3.3

Find the components of the definition.

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

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

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

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

Step 4

Plug in the components.

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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Step 5.1.3.1

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

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

Step 5.1.3.2

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

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

Step 5.1.3.3

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

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

Step 5.1.3.4

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

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

Step 5.1.3.5

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

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

Step 5.2

Reduce the expression by cancelling the common factors.

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Step 5.2.1

Cancel the common factor of $h$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

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

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

Step 5.2.2

Simplify the expression.

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Step 5.2.2.1

Move $h$ .

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

Step 5.2.2.2

Move $h^{2}$ .

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

Step 5.2.2.3

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

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

Step 6