Precalculus Examples

$f (x) = x + x^{4}$

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.

$f (x + h) = (x + h) + {(x + h)}^{4}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Use the Binomial Theorem.

$f (x + h) = x + h + x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4}$

Step 2.1.2.2

The final answer is $x + h + x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4}$ .

$x + h + x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4}$

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{3}$ .

$x + h + x^{4} + 4 h x^{3} + 6 x^{2} h^{2} + 4 x h^{3} + h^{4}$

Step 2.2.2

Move $x^{2}$ .

$x + h + x^{4} + 4 h x^{3} + 6 h^{2} x^{2} + 4 x h^{3} + h^{4}$

Step 2.2.3

Move $x$ .

$x + h + x^{4} + 4 h x^{3} + 6 h^{2} x^{2} + 4 h^{3} x + h^{4}$

Step 2.2.4

Move $x$ .

$h + x^{4} + 4 h x^{3} + 6 h^{2} x^{2} + 4 h^{3} x + h^{4} + x$

Step 2.2.5

Move $h$ .

$x^{4} + 4 h x^{3} + 6 h^{2} x^{2} + 4 h^{3} x + h^{4} + h + x$

Step 2.2.6

Move $x^{4}$ .

$4 h x^{3} + 6 h^{2} x^{2} + 4 h^{3} x + h^{4} + x^{4} + h + x$

Step 2.2.7

Move $4 h x^{3}$ .

$6 h^{2} x^{2} + 4 h^{3} x + h^{4} + 4 h x^{3} + x^{4} + h + x$

Step 2.2.8

Move $6 h^{2} x^{2}$ .

$4 h^{3} x + h^{4} + 6 h^{2} x^{2} + 4 h x^{3} + x^{4} + h + x$

Step 2.2.9

Reorder $4 h^{3} x$ and $h^{4}$ .

$h^{4} + 4 h^{3} x + 6 h^{2} x^{2} + 4 h x^{3} + x^{4} + h + x$

Step 2.3

Find the components of the definition.

$f (x + h) = h^{4} + 4 h^{3} x + 6 h^{2} x^{2} + 4 h x^{3} + x^{4} + h + x$

$f (x) = x + x^{4}$

$f (x + h) = h^{4} + 4 h^{3} x + 6 h^{2} x^{2} + 4 h x^{3} + x^{4} + h + x$

$f (x) = x + x^{4}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{h^{4} + 4 h^{3} x + 6 h^{2} x^{2} + 4 h x^{3} + x^{4} + h + x - (x + x^{4})}{h}$

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.

$\frac{h^{4} + 4 h^{3} x + 6 h^{2} x^{2} + 4 h x^{3} + x^{4} + h + x - x - x^{4}}{h}$

Step 4.1.2

Subtract $x^{4}$ from $x^{4}$ .

$\frac{h^{4} + 4 h^{3} x + 6 h^{2} x^{2} + 4 h x^{3} + h + x - x + 0}{h}$

Step 4.1.3

Add $h^{4} + 4 h^{3} x + 6 h^{2} x^{2} + 4 h x^{3} + h + x - x$ and $0$ .

$\frac{h^{4} + 4 h^{3} x + 6 h^{2} x^{2} + 4 h x^{3} + h + x - x}{h}$

Step 4.1.4

Subtract $x$ from $x$ .

$\frac{h^{4} + 4 h^{3} x + 6 h^{2} x^{2} + 4 h x^{3} + h + 0}{h}$

Step 4.1.5

Add $h^{4} + 4 h^{3} x + 6 h^{2} x^{2} + 4 h x^{3} + h$ and $0$ .

$\frac{h^{4} + 4 h^{3} x + 6 h^{2} x^{2} + 4 h x^{3} + h}{h}$

Step 4.1.6

Factor $h$ out of $h^{4} + 4 h^{3} x + 6 h^{2} x^{2} + 4 h x^{3} + h$ .

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Step 4.1.6.1

Factor $h$ out of $h^{4}$ .

$\frac{h \cdot h^{3} + 4 h^{3} x + 6 h^{2} x^{2} + 4 h x^{3} + h}{h}$

Step 4.1.6.2

Factor $h$ out of $4 h^{3} x$ .

$\frac{h (h^{3}) + h (4 h^{2} x) + 6 h^{2} x^{2} + 4 h x^{3} + h}{h}$

Step 4.1.6.3

Factor $h$ out of $6 h^{2} x^{2}$ .

$\frac{h (h^{3}) + h (4 h^{2} x) + h (6 h x^{2}) + 4 h x^{3} + h}{h}$

Step 4.1.6.4

Factor $h$ out of $4 h x^{3}$ .

$\frac{h (h^{3}) + h (4 h^{2} x) + h (6 h x^{2}) + h (4 x^{3}) + h}{h}$

Step 4.1.6.5

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

$\frac{h (h^{3}) + h (4 h^{2} x) + h (6 h x^{2}) + h (4 x^{3}) + h^{1}}{h}$

Step 4.1.6.6

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

$\frac{h (h^{3}) + h (4 h^{2} x) + h (6 h x^{2}) + h (4 x^{3}) + h \cdot 1}{h}$

Step 4.1.6.7

Factor $h$ out of $h (h^{3}) + h (4 h^{2} x)$ .

$\frac{h (h^{3} + 4 h^{2} x) + h (6 h x^{2}) + h (4 x^{3}) + h \cdot 1}{h}$

Step 4.1.6.8

Factor $h$ out of $h (h^{3} + 4 h^{2} x) + h (6 h x^{2})$ .

$\frac{h (h^{3} + 4 h^{2} x + 6 h x^{2}) + h (4 x^{3}) + h \cdot 1}{h}$

Step 4.1.6.9

Factor $h$ out of $h (h^{3} + 4 h^{2} x + 6 h x^{2}) + h (4 x^{3})$ .

$\frac{h (h^{3} + 4 h^{2} x + 6 h x^{2} + 4 x^{3}) + h \cdot 1}{h}$

Step 4.1.6.10

Factor $h$ out of $h (h^{3} + 4 h^{2} x + 6 h x^{2} + 4 x^{3}) + h \cdot 1$ .

$\frac{h (h^{3} + 4 h^{2} x + 6 h x^{2} + 4 x^{3} + 1)}{h}$

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.

$\frac{h (h^{3} + 4 h^{2} x + 6 h x^{2} + 4 x^{3} + 1)}{h}$

Step 4.2.1.2

Divide $h^{3} + 4 h^{2} x + 6 h x^{2} + 4 x^{3} + 1$ by $1$ .

$h^{3} + 4 h^{2} x + 6 h x^{2} + 4 x^{3} + 1$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Move $h^{2}$ .

$h^{3} + 4 x h^{2} + 6 h x^{2} + 4 x^{3} + 1$

Step 4.2.2.2

Move $h$ .

$h^{3} + 4 x h^{2} + 6 x^{2} h + 4 x^{3} + 1$

Step 4.2.2.3

Move $h^{3}$ .

$4 x h^{2} + 6 x^{2} h + 4 x^{3} + h^{3} + 1$

Step 4.2.2.4

Move $4 x h^{2}$ .

$6 x^{2} h + 4 x^{3} + 4 x h^{2} + h^{3} + 1$

Step 4.2.2.5

Reorder $6 x^{2} h$ and $4 x^{3}$ .

$4 x^{3} + 6 x^{2} h + 4 x h^{2} + h^{3} + 1$

Step 5