Calculus Examples

$f (6) = \frac{8 {(x + h)}^{2} - f {(8 x)}^{2}}{h}$

Step 1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \frac{8 {((6) + h)}^{2} - f {(8 (6))}^{2}}{h}$

Step 2

Simplify the result.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Rewrite ${(6 + h)}^{2}$ as $(6 + h) (6 + h)$ .

$f (6) = \frac{8 ((6 + h) (6 + h)) - f {(8 (6))}^{2}}{h}$

Step 2.1.2

Expand $(6 + h) (6 + h)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$f (6) = \frac{8 (6 (6 + h) + h (6 + h)) - f {(8 (6))}^{2}}{h}$

Step 2.1.2.2

Apply the distributive property.

$f (6) = \frac{8 (6 \cdot 6 + 6 h + h (6 + h)) - f {(8 (6))}^{2}}{h}$

Step 2.1.2.3

Apply the distributive property.

$f (6) = \frac{8 (6 \cdot 6 + 6 h + h \cdot 6 + h \cdot h) - f {(8 (6))}^{2}}{h}$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $6$ by $6$ .

$f (6) = \frac{8 (36 + 6 h + h \cdot 6 + h \cdot h) - f {(8 (6))}^{2}}{h}$

Step 2.1.3.1.2

Move $6$ to the left of $h$ .

$f (6) = \frac{8 (36 + 6 h + 6 \cdot h + h \cdot h) - f {(8 (6))}^{2}}{h}$

Step 2.1.3.1.3

Multiply $h$ by $h$ .

$f (6) = \frac{8 (36 + 6 h + 6 h + h^{2}) - f {(8 (6))}^{2}}{h}$

Step 2.1.3.2

Add $6 h$ and $6 h$ .

$f (6) = \frac{8 (36 + 12 h + h^{2}) - f {(8 (6))}^{2}}{h}$

Step 2.1.4

Apply the distributive property.

$f (6) = \frac{8 \cdot 36 + 8 (12 h) + 8 h^{2} - f {(8 (6))}^{2}}{h}$

Step 2.1.5

Simplify.

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Step 2.1.5.1

Multiply $8$ by $36$ .

$f (6) = \frac{288 + 8 (12 h) + 8 h^{2} - f {(8 (6))}^{2}}{h}$

Step 2.1.5.2

Multiply $12$ by $8$ .

$f (6) = \frac{288 + 96 h + 8 h^{2} - f {(8 (6))}^{2}}{h}$

Step 2.1.6

Multiply $8$ by $6$ .

$f (6) = \frac{288 + 96 h + 8 h^{2} - f \cdot 48^{2}}{h}$

Step 2.1.7

Raise $48$ to the power of $2$ .

$f (6) = \frac{288 + 96 h + 8 h^{2} - f \cdot 2304}{h}$

Step 2.1.8

Multiply $2304$ by $- 1$ .

$f (6) = \frac{288 + 96 h + 8 h^{2} - 2304 f}{h}$

Step 2.1.9

Factor $8$ out of $288 + 96 h + 8 h^{2} - 2304 f$ .

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Step 2.1.9.1

Factor $8$ out of $288$ .

$f (6) = \frac{8 \cdot 36 + 96 h + 8 h^{2} - 2304 f}{h}$

Step 2.1.9.2

Factor $8$ out of $96 h$ .

$f (6) = \frac{8 \cdot 36 + 8 (12 h) + 8 h^{2} - 2304 f}{h}$

Step 2.1.9.3

Factor $8$ out of $- 2304 f$ .

$f (6) = \frac{8 \cdot 36 + 8 (12 h) + 8 h^{2} + 8 (- 288 f)}{h}$

Step 2.1.9.4

Factor $8$ out of $8 \cdot 36 + 8 (12 h)$ .

$f (6) = \frac{8 \cdot (36 + 12 h) + 8 h^{2} + 8 (- 288 f)}{h}$

Step 2.1.9.5

Factor $8$ out of $8 \cdot (36 + 12 h) + 8 h^{2}$ .

$f (6) = \frac{8 \cdot (36 + 12 h + h^{2}) + 8 (- 288 f)}{h}$

Step 2.1.9.6

Factor $8$ out of $8 \cdot (36 + 12 h + h^{2}) + 8 (- 288 f)$ .

$f (6) = \frac{8 (36 + 12 h + h^{2} - 288 f)}{h}$

Step 2.2

The final answer is $\frac{8 (36 + 12 h + h^{2} - 288 f)}{h}$ .

$\frac{8 (36 + 12 h + h^{2} - 288 f)}{h}$

Step 3