Calculus Examples

f (x) = x^{2} - 6

$f (x) = x^{2} - 6$

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) = {(x + h)}^{2} - 6

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

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) = (x + h) (x + h) - 6

$f (x + h) = (x + h) (x + h) - 6$

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) = x (x + h) + h (x + h) - 6

$f (x + h) = x (x + h) + h (x + h) - 6$

Step 2.1.2.1.2.2

Apply the distributive property.

f (x + h) = x \cdot x + x h + h (x + h) - 6

$f (x + h) = x \cdot x + x h + h (x + h) - 6$

Step 2.1.2.1.2.3

Apply the distributive property.

f (x + h) = x \cdot x + x h + h x + h \cdot h - 6

$f (x + h) = x \cdot x + x h + h x + h \cdot h - 6$

f (x + h) = x \cdot x + x h + h x + h \cdot h - 6

$f (x + h) = x \cdot x + x h + h x + h \cdot h - 6$

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) = x^{2} + x h + h x + h \cdot h - 6

$f (x + h) = x^{2} + x h + h x + h \cdot h - 6$

Step 2.1.2.1.3.1.2

Multiply

h

$h$ by

h

$h$ .

f (x + h) = x^{2} + x h + h x + h^{2} - 6

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

f (x + h) = x^{2} + x h + h x + h^{2} - 6

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

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) = x^{2} + h x + h x + h^{2} - 6

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

Step 2.1.2.1.3.2.2

Add

h x

$h x$ and

h x

$h x$ .

f (x + h) = x^{2} + 2 h x + h^{2} - 6

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

f (x + h) = x^{2} + 2 h x + h^{2} - 6

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

f (x + h) = x^{2} + 2 h x + h^{2} - 6

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

f (x + h) = x^{2} + 2 h x + h^{2} - 6

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

Step 2.1.2.2

The final answer is

x^{2} + 2 h x + h^{2} - 6

$x^{2} + 2 h x + h^{2} - 6$ .

x^{2} + 2 h x + h^{2} - 6

$x^{2} + 2 h x + h^{2} - 6$

x^{2} + 2 h x + h^{2} - 6

$x^{2} + 2 h x + h^{2} - 6$

x^{2} + 2 h x + h^{2} - 6

$x^{2} + 2 h x + h^{2} - 6$

Step 2.2

Reorder.

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Step 2.2.1

Move

x^{2}

$x^{2}$ .

2 h x + h^{2} + x^{2} - 6

$2 h x + h^{2} + x^{2} - 6$

Step 2.2.2

Reorder

2 h x

$2 h x$ and

h^{2}

$h^{2}$ .

h^{2} + 2 h x + x^{2} - 6

$h^{2} + 2 h x + x^{2} - 6$

h^{2} + 2 h x + x^{2} - 6

$h^{2} + 2 h x + x^{2} - 6$

Step 2.3

Find the components of the definition.

f (x + h) = h^{2} + 2 h x + x^{2} - 6

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

f (x) = x^{2} - 6

$f (x) = x^{2} - 6$

f (x + h) = h^{2} + 2 h x + x^{2} - 6

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

f (x) = x^{2} - 6

$f (x) = x^{2} - 6$

Step 3

Plug in the components.

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

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

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^{2} + 2 h x + x^{2} - 6 - x^{2} - - 6}{h}

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

Step 4.1.2

Multiply

- 1

$- 1$ by

- 6

$- 6$ .

\frac{h^{2} + 2 h x + x^{2} - 6 - x^{2} + 6}{h}

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

Step 4.1.3

Subtract

x^{2}

$x^{2}$ from

x^{2}

$x^{2}$ .

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

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

Step 4.1.4

Add

h^{2}

$h^{2}$ and

0

$0$ .

\frac{h^{2} + 2 h x - 6 + 6}{h}

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

Step 4.1.5

Add

- 6

$- 6$ and

6

$6$ .

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

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

Step 4.1.6

Add

h^{2} + 2 h x

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

0

$0$ .

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

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

Step 4.1.7

Factor

h

$h$ out of

h^{2} + 2 h x

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

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Step 4.1.7.1

Factor

h

$h$ out of

h^{2}

$h^{2}$ .

\frac{h \cdot h + 2 h x}{h}

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

Step 4.1.7.2

Factor

h

$h$ out of

2 h x

$2 h x$ .

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

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

Step 4.1.7.3

Factor

h

$h$ out of

h (h) + h (2 x)

$h (h) + h (2 x)$ .

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

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

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

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

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

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

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

$h$ .

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Step 4.2.1.1

Cancel the common factor.

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

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

Step 4.2.1.2

Divide

h + 2 x

$h + 2 x$ by

1

$1$ .

h + 2 x

$h + 2 x$

h + 2 x

$h + 2 x$

Step 4.2.2

Reorder

h

$h$ and

2 x

$2 x$ .

2 x + h

$2 x + h$

2 x + h

$2 x + h$

2 x + h

$2 x + h$

Step 5

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