Calculus Examples

f (x) = 12 x^{2} - 3

$f (x) = 12 x^{2} - 3$

Step 1

Consider the difference quotient formula.

\frac{f (x + h) - f (x)}{h}

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

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at

x = x + h

$x = x + h$ .

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Step 2.1.1

Replace the variable

x

$x$ with

x + h

$x + h$ in the expression.

f (x + h) = 12 {(x + h)}^{2} - 3

$f (x + h) = 12 {(x + h)}^{2} - 3$

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

Rewrite

{(x + h)}^{2}

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

(x + h) (x + h)

$(x + h) (x + h)$ .

f (x + h) = 12 ((x + h) (x + h)) - 3

$f (x + h) = 12 ((x + h) (x + h)) - 3$

Step 2.1.2.1.2

Expand

(x + h) (x + h)

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

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Step 2.1.2.1.2.1

Apply the distributive property.

f (x + h) = 12 (x (x + h) + h (x + h)) - 3

$f (x + h) = 12 (x (x + h) + h (x + h)) - 3$

Step 2.1.2.1.2.2

Apply the distributive property.

f (x + h) = 12 (x \cdot x + x h + h (x + h)) - 3

$f (x + h) = 12 (x \cdot x + x h + h (x + h)) - 3$

Step 2.1.2.1.2.3

Apply the distributive property.

f (x + h) = 12 (x \cdot x + x h + h x + h \cdot h) - 3

$f (x + h) = 12 (x \cdot x + x h + h x + h \cdot h) - 3$

f (x + h) = 12 (x \cdot x + x h + h x + h \cdot h) - 3

$f (x + h) = 12 (x \cdot x + x h + h x + h \cdot h) - 3$

Step 2.1.2.1.3

Simplify and combine like terms.

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Step 2.1.2.1.3.1

Simplify each term.

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Step 2.1.2.1.3.1.1

Multiply

x

$x$ by

x

$x$ .

f (x + h) = 12 (x^{2} + x h + h x + h \cdot h) - 3

$f (x + h) = 12 (x^{2} + x h + h x + h \cdot h) - 3$

Step 2.1.2.1.3.1.2

Multiply

h

$h$ by

h

$h$ .

f (x + h) = 12 (x^{2} + x h + h x + h^{2}) - 3

$f (x + h) = 12 (x^{2} + x h + h x + h^{2}) - 3$

f (x + h) = 12 (x^{2} + x h + h x + h^{2}) - 3

$f (x + h) = 12 (x^{2} + x h + h x + h^{2}) - 3$

Step 2.1.2.1.3.2

Add

x h

$x h$ and

h x

$h x$ .

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Step 2.1.2.1.3.2.1

Reorder

x

$x$ and

h

$h$ .

f (x + h) = 12 (x^{2} + h x + h x + h^{2}) - 3

$f (x + h) = 12 (x^{2} + h x + h x + h^{2}) - 3$

Step 2.1.2.1.3.2.2

Add

h x

$h x$ and

h x

$h x$ .

f (x + h) = 12 (x^{2} + 2 h x + h^{2}) - 3

$f (x + h) = 12 (x^{2} + 2 h x + h^{2}) - 3$

f (x + h) = 12 (x^{2} + 2 h x + h^{2}) - 3

$f (x + h) = 12 (x^{2} + 2 h x + h^{2}) - 3$

f (x + h) = 12 (x^{2} + 2 h x + h^{2}) - 3

$f (x + h) = 12 (x^{2} + 2 h x + h^{2}) - 3$

Step 2.1.2.1.4

Apply the distributive property.

f (x + h) = 12 x^{2} + 12 (2 h x) + 12 h^{2} - 3

$f (x + h) = 12 x^{2} + 12 (2 h x) + 12 h^{2} - 3$

Step 2.1.2.1.5

Multiply

2

$2$ by

12

$12$ .

f (x + h) = 12 x^{2} + 24 h x + 12 h^{2} - 3

$f (x + h) = 12 x^{2} + 24 h x + 12 h^{2} - 3$

f (x + h) = 12 x^{2} + 24 h x + 12 h^{2} - 3

$f (x + h) = 12 x^{2} + 24 h x + 12 h^{2} - 3$

Step 2.1.2.2

The final answer is

12 x^{2} + 24 h x + 12 h^{2} - 3

$12 x^{2} + 24 h x + 12 h^{2} - 3$ .

12 x^{2} + 24 h x + 12 h^{2} - 3

$12 x^{2} + 24 h x + 12 h^{2} - 3$

12 x^{2} + 24 h x + 12 h^{2} - 3

$12 x^{2} + 24 h x + 12 h^{2} - 3$

12 x^{2} + 24 h x + 12 h^{2} - 3

$12 x^{2} + 24 h x + 12 h^{2} - 3$

Step 2.2

Reorder.

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Step 2.2.1

Move

12 x^{2}

$12 x^{2}$ .

24 h x + 12 h^{2} + 12 x^{2} - 3

$24 h x + 12 h^{2} + 12 x^{2} - 3$

Step 2.2.2

Reorder

24 h x

$24 h x$ and

12 h^{2}

$12 h^{2}$ .

12 h^{2} + 24 h x + 12 x^{2} - 3

$12 h^{2} + 24 h x + 12 x^{2} - 3$

12 h^{2} + 24 h x + 12 x^{2} - 3

$12 h^{2} + 24 h x + 12 x^{2} - 3$

Step 2.3

Find the components of the definition.

f (x + h) = 12 h^{2} + 24 h x + 12 x^{2} - 3

$f (x + h) = 12 h^{2} + 24 h x + 12 x^{2} - 3$

f (x) = 12 x^{2} - 3

$f (x) = 12 x^{2} - 3$

f (x + h) = 12 h^{2} + 24 h x + 12 x^{2} - 3

$f (x + h) = 12 h^{2} + 24 h x + 12 x^{2} - 3$

f (x) = 12 x^{2} - 3

$f (x) = 12 x^{2} - 3$

Step 3

Plug in the components.

\frac{f (x + h) - f (x)}{h} = \frac{12 h^{2} + 24 h x + 12 x^{2} - 3 - (12 x^{2} - 3)}{h}

$\frac{f (x + h) - f (x)}{h} = \frac{12 h^{2} + 24 h x + 12 x^{2} - 3 - (12 x^{2} - 3)}{h}$

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{12 h^{2} + 24 h x + 12 x^{2} - 3 - (12 x^{2}) - - 3}{h}

$\frac{12 h^{2} + 24 h x + 12 x^{2} - 3 - (12 x^{2}) - - 3}{h}$

Step 4.1.2

Multiply

12

$12$ by

- 1

$- 1$ .

\frac{12 h^{2} + 24 h x + 12 x^{2} - 3 - 12 x^{2} - - 3}{h}

$\frac{12 h^{2} + 24 h x + 12 x^{2} - 3 - 12 x^{2} - - 3}{h}$

Step 4.1.3

Multiply

- 1

$- 1$ by

- 3

$- 3$ .

\frac{12 h^{2} + 24 h x + 12 x^{2} - 3 - 12 x^{2} + 3}{h}

$\frac{12 h^{2} + 24 h x + 12 x^{2} - 3 - 12 x^{2} + 3}{h}$

Step 4.1.4

Subtract

12 x^{2}

$12 x^{2}$ from

12 x^{2}

$12 x^{2}$ .

\frac{12 h^{2} + 24 h x + 0 - 3 + 3}{h}

$\frac{12 h^{2} + 24 h x + 0 - 3 + 3}{h}$

Step 4.1.5

Add

12 h^{2}

$12 h^{2}$ and

0

$0$ .

\frac{12 h^{2} + 24 h x - 3 + 3}{h}

$\frac{12 h^{2} + 24 h x - 3 + 3}{h}$

Step 4.1.6

Add

- 3

$- 3$ and

3

$3$ .

\frac{12 h^{2} + 24 h x + 0}{h}

$\frac{12 h^{2} + 24 h x + 0}{h}$

Step 4.1.7

Add

12 h^{2} + 24 h x

$12 h^{2} + 24 h x$ and

0

$0$ .

\frac{12 h^{2} + 24 h x}{h}

$\frac{12 h^{2} + 24 h x}{h}$

Step 4.1.8

Factor

12 h

$12 h$ out of

12 h^{2} + 24 h x

$12 h^{2} + 24 h x$ .

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Step 4.1.8.1

Factor

12 h

$12 h$ out of

12 h^{2}

$12 h^{2}$ .

\frac{12 h \cdot h + 24 h x}{h}

$\frac{12 h \cdot h + 24 h x}{h}$

Step 4.1.8.2

Factor

12 h

$12 h$ out of

24 h x

$24 h x$ .

\frac{12 h \cdot h + 12 h (2 x)}{h}

$\frac{12 h \cdot h + 12 h (2 x)}{h}$

Step 4.1.8.3

Factor

12 h

$12 h$ out of

12 h \cdot h + 12 h (2 x)

$12 h \cdot h + 12 h (2 x)$ .

\frac{12 h (h + 2 x)}{h}

$\frac{12 h (h + 2 x)}{h}$

\frac{12 h (h + 2 x)}{h}

$\frac{12 h (h + 2 x)}{h}$

\frac{12 h (h + 2 x)}{h}

$\frac{12 h (h + 2 x)}{h}$

Step 4.2

Simplify terms.

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Step 4.2.1

Cancel the common factor of

h

$h$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{12 h (h + 2 x)}{h}$

Step 4.2.1.2

Divide

$12 (h + 2 x)$ by

$1$ .

$12 (h + 2 x)$

Step 4.2.2

Apply the distributive property.

$12 h + 12 (2 x)$

Step 4.2.3

Simplify the expression.

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Step 4.2.3.1

Multiply

$2$ by

$12$ .

$12 h + 24 x$

Step 4.2.3.2

Reorder

$12 h$ and

$24 x$ .

$24 x + 12 h$

Step 5

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