Calculus Examples

$f (x) = x^{2} - 4$ , $f (- 3 + h) - f \cdot - 3$

Step 1

Replace the variable $x$ with $- 3 + h$ in the expression.

$f (- 3 + h) = {(- 3 + h)}^{2} - 4$

Step 2

Replace the function designators in $f (- 3 + h) - f \cdot - 3$ with the actual functions.

$({(- 3 + h)}^{2} - 4) - (x^{2} - 4) \cdot - 3$

Step 3

Simplify $({(- 3 + h)}^{2} - 4) - (x^{2} - 4) \cdot - 3$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite ${(- 3 + h)}^{2}$ as $(- 3 + h) (- 3 + h)$ .

$(- 3 + h) (- 3 + h) - 4 - (x^{2} - 4) \cdot - 3$

Step 3.1.2

Expand $(- 3 + h) (- 3 + h)$ using the FOIL Method.

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Step 3.1.2.1

Apply the distributive property.

$- 3 (- 3 + h) + h (- 3 + h) - 4 - (x^{2} - 4) \cdot - 3$

Step 3.1.2.2

Apply the distributive property.

$- 3 \cdot - 3 - 3 h + h (- 3 + h) - 4 - (x^{2} - 4) \cdot - 3$

Step 3.1.2.3

Apply the distributive property.

$- 3 \cdot - 3 - 3 h + h \cdot - 3 + h \cdot h - 4 - (x^{2} - 4) \cdot - 3$

Step 3.1.3

Simplify and combine like terms.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Multiply $- 3$ by $- 3$ .

$9 - 3 h + h \cdot - 3 + h \cdot h - 4 - (x^{2} - 4) \cdot - 3$

Step 3.1.3.1.2

Move $- 3$ to the left of $h$ .

$9 - 3 h - 3 \cdot h + h \cdot h - 4 - (x^{2} - 4) \cdot - 3$

Step 3.1.3.1.3

Multiply $h$ by $h$ .

$9 - 3 h - 3 h + h^{2} - 4 - (x^{2} - 4) \cdot - 3$

Step 3.1.3.2

Subtract $3 h$ from $- 3 h$ .

$9 - 6 h + h^{2} - 4 - (x^{2} - 4) \cdot - 3$

Step 3.1.4

Apply the distributive property.

$9 - 6 h + h^{2} - 4 + (- x^{2} - - 4) \cdot - 3$

Step 3.1.5

Multiply $- 1$ by $- 4$ .

$9 - 6 h + h^{2} - 4 + (- x^{2} + 4) \cdot - 3$

Step 3.1.6

Apply the distributive property.

$9 - 6 h + h^{2} - 4 - x^{2} \cdot - 3 + 4 \cdot - 3$

Step 3.1.7

Multiply $- 3$ by $- 1$ .

$9 - 6 h + h^{2} - 4 + 3 x^{2} + 4 \cdot - 3$

Step 3.1.8

Multiply $4$ by $- 3$ .

$9 - 6 h + h^{2} - 4 + 3 x^{2} - 12$

Step 3.2

Simplify the expression.

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Step 3.2.1

Subtract $4$ from $9$ .

$- 6 h + h^{2} + 5 + 3 x^{2} - 12$

Step 3.2.2

Subtract $12$ from $5$ .

$- 6 h + h^{2} + 3 x^{2} - 7$

Step 3.2.3

Move $- 6 h$ .

$h^{2} + 3 x^{2} - 6 h - 7$