Calculus Examples

$f (x) = x^{5}$

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.

$f (x + h) = {(x + h)}^{5}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Use the Binomial Theorem.

$f (x + h) = x^{5} + 5 x^{4} h + 10 x^{3} h^{2} + 10 x^{2} h^{3} + 5 x h^{4} + h^{5}$

Step 2.1.2.2

The final answer is $x^{5} + 5 x^{4} h + 10 x^{3} h^{2} + 10 x^{2} h^{3} + 5 x h^{4} + h^{5}$ .

$x^{5} + 5 x^{4} h + 10 x^{3} h^{2} + 10 x^{2} h^{3} + 5 x h^{4} + h^{5}$

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{4}$ .

$x^{5} + 5 h x^{4} + 10 x^{3} h^{2} + 10 x^{2} h^{3} + 5 x h^{4} + h^{5}$

Step 2.2.2

Move $x^{3}$ .

$x^{5} + 5 h x^{4} + 10 h^{2} x^{3} + 10 x^{2} h^{3} + 5 x h^{4} + h^{5}$

Step 2.2.3

Move $x^{2}$ .

$x^{5} + 5 h x^{4} + 10 h^{2} x^{3} + 10 h^{3} x^{2} + 5 x h^{4} + h^{5}$

Step 2.2.4

Move $x$ .

$x^{5} + 5 h x^{4} + 10 h^{2} x^{3} + 10 h^{3} x^{2} + 5 h^{4} x + h^{5}$

Step 2.2.5

Move $x^{5}$ .

$5 h x^{4} + 10 h^{2} x^{3} + 10 h^{3} x^{2} + 5 h^{4} x + h^{5} + x^{5}$

Step 2.2.6

Move $5 h x^{4}$ .

$10 h^{2} x^{3} + 10 h^{3} x^{2} + 5 h^{4} x + h^{5} + 5 h x^{4} + x^{5}$

Step 2.2.7

Move $10 h^{2} x^{3}$ .

$10 h^{3} x^{2} + 5 h^{4} x + h^{5} + 10 h^{2} x^{3} + 5 h x^{4} + x^{5}$

Step 2.2.8

Move $10 h^{3} x^{2}$ .

$5 h^{4} x + h^{5} + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4} + x^{5}$

Step 2.2.9

Reorder $5 h^{4} x$ and $h^{5}$ .

$h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4} + x^{5}$

Step 2.3

Find the components of the definition.

$f (x + h) = h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4} + x^{5}$

$f (x) = x^{5}$

$f (x + h) = h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4} + x^{5}$

$f (x) = x^{5}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4} + x^{5} - (x^{5})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Subtract $x^{5}$ from $x^{5}$ .

$\frac{h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4} + 0}{h}$

Step 4.1.2

Add $h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4}$ and $0$ .

$\frac{h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4}}{h}$

Step 4.1.3

Factor $h$ out of $h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4}$ .

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Step 4.1.3.1

Factor $h$ out of $h^{5}$ .

$\frac{h \cdot h^{4} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4}}{h}$

Step 4.1.3.2

Factor $h$ out of $5 h^{4} x$ .

$\frac{h (h^{4}) + h (5 h^{3} x) + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4}}{h}$

Step 4.1.3.3

Factor $h$ out of $10 h^{3} x^{2}$ .

$\frac{h (h^{4}) + h (5 h^{3} x) + h (10 h^{2} x^{2}) + 10 h^{2} x^{3} + 5 h x^{4}}{h}$

Step 4.1.3.4

Factor $h$ out of $10 h^{2} x^{3}$ .

$\frac{h (h^{4}) + h (5 h^{3} x) + h (10 h^{2} x^{2}) + h (10 h x^{3}) + 5 h x^{4}}{h}$

Step 4.1.3.5

Factor $h$ out of $5 h x^{4}$ .

$\frac{h (h^{4}) + h (5 h^{3} x) + h (10 h^{2} x^{2}) + h (10 h x^{3}) + h (5 x^{4})}{h}$

Step 4.1.3.6

Factor $h$ out of $h (h^{4}) + h (5 h^{3} x)$ .

$\frac{h (h^{4} + 5 h^{3} x) + h (10 h^{2} x^{2}) + h (10 h x^{3}) + h (5 x^{4})}{h}$

Step 4.1.3.7

Factor $h$ out of $h (h^{4} + 5 h^{3} x) + h (10 h^{2} x^{2})$ .

$\frac{h (h^{4} + 5 h^{3} x + 10 h^{2} x^{2}) + h (10 h x^{3}) + h (5 x^{4})}{h}$

Step 4.1.3.8

Factor $h$ out of $h (h^{4} + 5 h^{3} x + 10 h^{2} x^{2}) + h (10 h x^{3})$ .

$\frac{h (h^{4} + 5 h^{3} x + 10 h^{2} x^{2} + 10 h x^{3}) + h (5 x^{4})}{h}$

Step 4.1.3.9

Factor $h$ out of $h (h^{4} + 5 h^{3} x + 10 h^{2} x^{2} + 10 h x^{3}) + h (5 x^{4})$ .

$\frac{h (h^{4} + 5 h^{3} x + 10 h^{2} x^{2} + 10 h x^{3} + 5 x^{4})}{h}$

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.

$\frac{h (h^{4} + 5 h^{3} x + 10 h^{2} x^{2} + 10 h x^{3} + 5 x^{4})}{h}$

Step 4.2.1.2

Divide $h^{4} + 5 h^{3} x + 10 h^{2} x^{2} + 10 h x^{3} + 5 x^{4}$ by $1$ .

$h^{4} + 5 h^{3} x + 10 h^{2} x^{2} + 10 h x^{3} + 5 x^{4}$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Move $h^{3}$ .

$h^{4} + 5 x h^{3} + 10 h^{2} x^{2} + 10 h x^{3} + 5 x^{4}$

Step 4.2.2.2

Move $h^{2}$ .

$h^{4} + 5 x h^{3} + 10 x^{2} h^{2} + 10 h x^{3} + 5 x^{4}$

Step 4.2.2.3

Move $h$ .

$h^{4} + 5 x h^{3} + 10 x^{2} h^{2} + 10 x^{3} h + 5 x^{4}$

Step 4.2.2.4

Move $h^{4}$ .

$5 x h^{3} + 10 x^{2} h^{2} + 10 x^{3} h + 5 x^{4} + h^{4}$

Step 4.2.2.5

Move $5 x h^{3}$ .

$10 x^{2} h^{2} + 10 x^{3} h + 5 x^{4} + 5 x h^{3} + h^{4}$

Step 4.2.2.6

Move $10 x^{2} h^{2}$ .

$10 x^{3} h + 5 x^{4} + 10 x^{2} h^{2} + 5 x h^{3} + h^{4}$

Step 4.2.2.7

Reorder $10 x^{3} h$ and $5 x^{4}$ .

$5 x^{4} + 10 x^{3} h + 10 x^{2} h^{2} + 5 x h^{3} + h^{4}$

Step 5