Evaluate the Difference Quotient f(x)=-x^3 , (f(a+h)-f(a))/h

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

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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

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

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

Simplify the result.

Use the Binomial Theorem.

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

Apply the distributive property.

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

Simplify.

Multiply $3$ by $- 1$ .

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

Multiply $3$ by $- 1$ .

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

Remove parentheses.

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

The final answer is $- x^{3} - 3 x^{2} h - 3 x h^{2} - h^{3}$ .

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

Reorder.

Move $x^{2}$ .

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

Move $x$ .

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

Move $- x^{3}$ .

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

Move $- 3 h x^{2}$ .

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

Reorder $- 3 h^{2} x$ and $- h^{3}$ .

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

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

Simplify the numerator.

Multiply $- - x^{3}$ .

Multiply $- 1$ by $- 1$ .

$\frac{- h^{3} - 3 h^{2} x - 3 h x^{2} - x^{3} + 1 x^{3}}{h}$

Multiply $x^{3}$ by $1$ .

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

Add $- x^{3}$ and $x^{3}$ .

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

Add $- h^{3} - 3 h^{2} x - 3 h x^{2}$ and $0$ .

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

Factor $h$ out of $- h^{3} - 3 h^{2} x - 3 h x^{2}$ .

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

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

Factor $h$ out of $- 3 h^{2} x$ .

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

Factor $h$ out of $- 3 h x^{2}$ .

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

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

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

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

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

Reduce the expression by cancelling the common factors.

Cancel the common factor of $h$ .

Cancel the common factor.

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

Divide $- h^{2} - 3 h x - 3 x^{2}$ by $1$ .

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

Simplify the expression.

Move $h$ .

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

Move $- h^{2}$ .

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

Reorder $- 3 x h$ and $- 3 x^{2}$ .

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

Step 5

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

$- 3 {(a)}^{2} - 3 a \cdot h - h^{2}$

Step 6

Multiply $- 3 a$ by $h$ .

$- 3 a^{2} - 3 a h - h^{2}$

Step 7

Evaluate the Difference Quotient f(x)=-x^3 , (f(a+h)-f(a))/h

Related Questions

Support LCMGCF.com by disabling your adblocker.