Calculus Examples

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

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.

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

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

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

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

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

Step 2.1.2.1.2

Expand

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

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Step 2.1.2.1.2.1

Apply the distributive property.

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

Step 2.1.2.1.2.2

Apply the distributive property.

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

Step 2.1.2.1.2.3

Apply the distributive property.

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

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

$x$ by

$x$ .

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

Step 2.1.2.1.3.1.2

Multiply

$h$ by

$h$ .

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

Step 2.1.2.1.3.2

Add

$x h$ and

$h x$ .

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Step 2.1.2.1.3.2.1

Reorder

$x$ and

$h$ .

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

Step 2.1.2.1.3.2.2

Add

$h x$ and

$h x$ .

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

Step 2.1.2.1.4

Apply the distributive property.

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

Step 2.1.2.2

The final answer is

$x^{2} + 2 h x + h^{2} + 3 x + 3 h$ .

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

Step 2.2

Reorder.

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Step 2.2.1

Move

$3 x$ .

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

Step 2.2.2

Move

$x^{2}$ .

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

Step 2.2.3

Reorder

$2 h x$ and

$h^{2}$ .

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

Step 2.3

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

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.

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

Step 4.1.2

Multiply

$3$ by

$- 1$ .

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

Step 4.1.3

Subtract

$x^{2}$ from

$x^{2}$ .

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

Step 4.1.4

Add

$h^{2}$ and

$0$ .

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

Step 4.1.5

Subtract

$3 x$ from

$3 x$ .

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

Step 4.1.6

Add

$h^{2} + 2 h x + 3 h$ and

$0$ .

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

Step 4.1.7

Factor

$h$ out of

$h^{2} + 2 h x + 3 h$ .

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Step 4.1.7.1

Factor

$h$ out of

$h^{2}$ .

$\frac{h \cdot h + 2 h x + 3 h}{h}$

Step 4.1.7.2

Factor

$h$ out of

$2 h x$ .

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

Step 4.1.7.3

Factor

$h$ out of

$3 h$ .

$\frac{h (h) + h (2 x) + h \cdot 3}{h}$

Step 4.1.7.4

Factor

$h$ out of

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

$\frac{h (h + 2 x) + h \cdot 3}{h}$

Step 4.1.7.5

Factor

$h$ out of

$h (h + 2 x) + h \cdot 3$ .

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

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.

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

Step 4.2.1.2

Divide

$h + 2 x + 3$ by

$1$ .

$h + 2 x + 3$

Step 4.2.2

Reorder

$h$ and

$2 x$ .

$2 x + h + 3$

Step 5

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