Find [f(x+h)-f(x)]/h f(x)=6x^3

$f (x) = 6 x^{3}$

Consider the difference quotient formula.

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

Find the components of the definition.

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Evaluate the function at $x = x + h$ .

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

$f (x + h) = 6 {(x + h)}^{3}$

Simplify the result.

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Use the Binomial Theorem.

$f (x + h) = 6 (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3})$

Apply the distributive property.

$f (x + h) = 6 x^{3} + 6 (3 x^{2} h) + 6 (3 x h^{2}) + 6 h^{3}$

Simplify.

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Multiply $3$ by $6$ .

$f (x + h) = 6 x^{3} + 18 (x^{2} h) + 6 (3 x h^{2}) + 6 h^{3}$

Multiply $3$ by $6$ .

$f (x + h) = 6 x^{3} + 18 (x^{2} h) + 18 (x h^{2}) + 6 h^{3}$

Remove parentheses.

$f (x + h) = 6 x^{3} + 18 x^{2} h + 18 x h^{2} + 6 h^{3}$

The final answer is $6 x^{3} + 18 x^{2} h + 18 x h^{2} + 6 h^{3}$ .

$6 x^{3} + 18 x^{2} h + 18 x h^{2} + 6 h^{3}$

Reorder.

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Move $x^{2}$ .

$6 x^{3} + 18 h x^{2} + 18 x h^{2} + 6 h^{3}$

Move $x$ .

$6 x^{3} + 18 h x^{2} + 18 h^{2} x + 6 h^{3}$

Move $6 x^{3}$ .

$18 h x^{2} + 18 h^{2} x + 6 h^{3} + 6 x^{3}$

Move $18 h x^{2}$ .

$18 h^{2} x + 6 h^{3} + 18 h x^{2} + 6 x^{3}$

Reorder $18 h^{2} x$ and $6 h^{3}$ .

$6 h^{3} + 18 h^{2} x + 18 h x^{2} + 6 x^{3}$

Find the components of the definition.

$f (x + h) = 6 h^{3} + 18 h^{2} x + 18 h x^{2} + 6 x^{3}$

$f (x) = 6 x^{3}$

$f (x + h) = 6 h^{3} + 18 h^{2} x + 18 h x^{2} + 6 x^{3}$

$f (x) = 6 x^{3}$

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{6 h^{3} + 18 h^{2} x + 18 h x^{2} + 6 x^{3} - (6 x^{3})}{h}$

Simplify.

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Simplify the numerator.

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Multiply $6$ by $- 1$ .

$\frac{6 h^{3} + 18 h^{2} x + 18 h x^{2} + 6 x^{3} - 6 x^{3}}{h}$

Subtract $6 x^{3}$ from $6 x^{3}$ .

$\frac{6 h^{3} + 18 h^{2} x + 18 h x^{2} + 0}{h}$

Add $6 h^{3} + 18 h^{2} x + 18 h x^{2}$ and $0$ .

$\frac{6 h^{3} + 18 h^{2} x + 18 h x^{2}}{h}$

Factor $6 h$ out of $6 h^{3} + 18 h^{2} x + 18 h x^{2}$ .

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Factor $6 h$ out of $6 h^{3}$ .

$\frac{6 h \cdot h^{2} + 18 h^{2} x + 18 h x^{2}}{h}$

Factor $6 h$ out of $18 h^{2} x$ .

$\frac{6 h \cdot h^{2} + 6 h (3 h x) + 18 h x^{2}}{h}$

Factor $6 h$ out of $18 h x^{2}$ .

$\frac{6 h \cdot h^{2} + 6 h (3 h x) + 6 h (3 x^{2})}{h}$

Factor $6 h$ out of $6 h \cdot h^{2} + 6 h (3 h x)$ .

$\frac{6 h (h^{2} + 3 h x) + 6 h (3 x^{2})}{h}$

Factor $6 h$ out of $6 h (h^{2} + 3 h x) + 6 h (3 x^{2})$ .

$\frac{6 h (h^{2} + 3 h x + 3 x^{2})}{h}$

Simplify terms.

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Cancel the common factor of $h$ .

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Cancel the common factor.

$\frac{6 h (h^{2} + 3 h x + 3 x^{2})}{h}$

Divide $6 (h^{2} + 3 h x + 3 x^{2})$ by $1$ .

$6 (h^{2} + 3 h x + 3 x^{2})$

Apply the distributive property.

$6 h^{2} + 6 (3 h x) + 6 (3 x^{2})$

Simplify.

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Multiply $3$ by $6$ .

$6 h^{2} + 18 (h x) + 6 (3 x^{2})$

Multiply $3$ by $6$ .

$6 h^{2} + 18 (h x) + 18 x^{2}$

Move $h$ .

$6 h^{2} + 18 x h + 18 x^{2}$

Move $6 h^{2}$ .

$18 x h + 18 x^{2} + 6 h^{2}$

Reorder $18 x h$ and $18 x^{2}$ .

$18 x^{2} + 18 x h + 6 h^{2}$

Find [f(x+h)-f(x)]/h f(x)=6x^3

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