Calculus Examples

$f (x) = 4 x^{2}$

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$f (x + h) = 4 ((x + h) (x + h))$

Step 2.1.2.2

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 2.1.2.2.1

Apply the distributive property.

$f (x + h) = 4 (x (x + h) + h (x + h))$

Step 2.1.2.2.2

Apply the distributive property.

$f (x + h) = 4 (x \cdot x + x h + h (x + h))$

Step 2.1.2.2.3

Apply the distributive property.

$f (x + h) = 4 (x \cdot x + x h + h x + h \cdot h)$

Step 2.1.2.3

Simplify and combine like terms.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Multiply $x$ by $x$ .

$f (x + h) = 4 (x^{2} + x h + h x + h \cdot h)$

Step 2.1.2.3.1.2

Multiply $h$ by $h$ .

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

Step 2.1.2.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.3.2.1

Reorder $x$ and $h$ .

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

Step 2.1.2.3.2.2

Add $h x$ and $h x$ .

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

Step 2.1.2.4

Apply the distributive property.

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

Step 2.1.2.5

Multiply $2$ by $4$ .

$f (x + h) = 4 x^{2} + 8 h x + 4 h^{2}$

Step 2.1.2.6

The final answer is $4 x^{2} + 8 h x + 4 h^{2}$ .

$4 x^{2} + 8 h x + 4 h^{2}$

Step 2.2

Reorder.

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Step 2.2.1

Move $4 x^{2}$ .

$8 h x + 4 h^{2} + 4 x^{2}$

Step 2.2.2

Reorder $8 h x$ and $4 h^{2}$ .

$4 h^{2} + 8 h x + 4 x^{2}$

Step 2.3

Find the components of the definition.

$f (x + h) = 4 h^{2} + 8 h x + 4 x^{2}$

$f (x) = 4 x^{2}$

$f (x + h) = 4 h^{2} + 8 h x + 4 x^{2}$

$f (x) = 4 x^{2}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{4 h^{2} + 8 h x + 4 x^{2} - (4 x^{2})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor using the perfect square rule.

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Step 4.1.1.1

Rewrite $4 h^{2}$ as ${(2 h)}^{2}$ .

$\frac{{(2 h)}^{2} + 8 h x + 4 x^{2} - 1 \cdot 4 x^{2}}{h}$

Step 4.1.1.2

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\frac{{(2 h)}^{2} + 8 h x + {(2 x)}^{2} - 1 \cdot 4 x^{2}}{h}$

Step 4.1.1.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 h x = 2 \cdot (2 h) \cdot (2 x)$

Step 4.1.1.4

Rewrite the polynomial.

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

Step 4.1.1.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 2 h$ and $b = 2 x$ .

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

Step 4.1.2

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

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

Step 4.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 h + 2 x$ and $b = 2 x$ .

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

Step 4.1.4

Simplify.

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Step 4.1.4.1

Add $2 x$ and $2 x$ .

$\frac{(2 h + 4 x) (2 h + 2 x - (2 x))}{h}$

Step 4.1.4.2

Factor $2$ out of $2 h + 4 x$ .

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Step 4.1.4.2.1

Factor $2$ out of $2 h$ .

$\frac{(2 h + 4 x) (2 h + 2 x - (2 x))}{h}$

Step 4.1.4.2.2

Factor $2$ out of $4 x$ .

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

Step 4.1.4.2.3

Factor $2$ out of $2 h + 2 (2 x)$ .

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

Step 4.1.4.3

Multiply $2$ by $- 1$ .

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

Step 4.1.5

Combine the opposite terms in $2 h + 2 x - 2 x$ .

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Step 4.1.5.1

Subtract $2 x$ from $2 x$ .

$\frac{2 (h + 2 x) (2 h + 0)}{h}$

Step 4.1.5.2

Add $2 h$ and $0$ .

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

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

Step 4.1.6

Multiply $2$ by $2$ .

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

Step 4.2

Simplify terms.

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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.

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

Step 4.2.1.2

Divide $4 (h + 2 x)$ by $1$ .

$4 (h + 2 x)$

Step 4.2.2

Apply the distributive property.

$4 h + 4 (2 x)$

Step 4.2.3

Simplify the expression.

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Step 4.2.3.1

Multiply $2$ by $4$ .

$4 h + 8 x$

Step 4.2.3.2

Reorder $4 h$ and $8 x$ .

$8 x + 4 h$

Step 5