Find the Difference Quotient f(x)=4x-3x^2+6

$f (x) = 4 x - 3 x^{2} + 6$

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.

$f (x + h) = 4 (x + h) - 3 {(x + h)}^{2} + 6$

Simplify the result.

Simplify each term.

Apply the distributive property.

$f (x + h) = 4 x + 4 h - 3 {(x + h)}^{2} + 6$

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

$f (x + h) = 4 x + 4 h - 3 ((x + h) (x + h)) + 6$

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

Apply the distributive property.

$f (x + h) = 4 x + 4 h - 3 (x (x + h) + h (x + h)) + 6$

Apply the distributive property.

$f (x + h) = 4 x + 4 h - 3 (x \cdot x + x h + h (x + h)) + 6$

Apply the distributive property.

$f (x + h) = 4 x + 4 h - 3 (x \cdot x + x h + h x + h \cdot h) + 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $h$ by $h$ .

$f (x + h) = 4 x + 4 h - 3 (x^{2} + x h + h x + h^{2}) + 6$

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

$f (x + h) = 4 x + 4 h - 3 (x^{2} + h x + h x + h^{2}) + 6$

Add $h x$ and $h x$ .

$f (x + h) = 4 x + 4 h - 3 (x^{2} + 2 h x + h^{2}) + 6$

Apply the distributive property.

$f (x + h) = 4 x + 4 h - 3 x^{2} - 3 (2 h x) - 3 h^{2} + 6$

Multiply $2$ by $- 3$ .

$f (x + h) = 4 x + 4 h - 3 x^{2} - 6 h x - 3 h^{2} + 6$

The final answer is $4 x + 4 h - 3 x^{2} - 6 h x - 3 h^{2} + 6$ .

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

Reorder.

Move $4 x$ .

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

Move $4 h$ .

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

Move $- 3 x^{2}$ .

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

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

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

Find the components of the definition.

$f (x + h) = - 3 h^{2} - 6 h x - 3 x^{2} + 4 h + 4 x + 6$

$f (x) = 4 x - 3 x^{2} + 6$

$f (x + h) = - 3 h^{2} - 6 h x - 3 x^{2} + 4 h + 4 x + 6$

$f (x) = 4 x - 3 x^{2} + 6$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- 3 h^{2} - 6 h x - 3 x^{2} + 4 h + 4 x + 6 - (4 x - 3 x^{2} + 6)}{h}$

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

$\frac{- 3 h^{2} - 6 h x - 3 x^{2} + 4 h + 4 x + 6 - (4 x) - (- 3 x^{2}) - 1 \cdot 6}{h}$

Simplify.

Multiply $4$ by $- 1$ .

$\frac{- 3 h^{2} - 6 h x - 3 x^{2} + 4 h + 4 x + 6 - 4 x - (- 3 x^{2}) - 1 \cdot 6}{h}$

Multiply $- 3$ by $- 1$ .

$\frac{- 3 h^{2} - 6 h x - 3 x^{2} + 4 h + 4 x + 6 - 4 x + 3 x^{2} - 1 \cdot 6}{h}$

Multiply $- 1$ by $6$ .

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

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

$\frac{- 3 h^{2} - 6 h x + 4 h + 4 x + 6 - 4 x + 0 - 6}{h}$

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

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

Subtract $4 x$ from $4 x$ .

$\frac{- 3 h^{2} - 6 h x + 4 h + 0 + 6 - 6}{h}$

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

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

Subtract $6$ from $6$ .

$\frac{- 3 h^{2} - 6 h x + 4 h + 0}{h}$

Add $- 3 h^{2} - 6 h x + 4 h$ and $0$ .

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

Factor $h$ out of $- 3 h^{2} - 6 h x + 4 h$ .

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

$\frac{h (- 3 h) - 6 h x + 4 h}{h}$

Factor $h$ out of $- 6 h x$ .

$\frac{h (- 3 h) + h (- 6 x) + 4 h}{h}$

Factor $h$ out of $4 h$ .

$\frac{h (- 3 h) + h (- 6 x) + h \cdot 4}{h}$

Factor $h$ out of $h (- 3 h) + h (- 6 x)$ .

$\frac{h (- 3 h - 6 x) + h \cdot 4}{h}$

Factor $h$ out of $h (- 3 h - 6 x) + h \cdot 4$ .

$\frac{h (- 3 h - 6 x + 4)}{h}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{h (- 3 h - 6 x + 4)}{h}$

Divide $- 3 h - 6 x + 4$ by $1$ .

$- 3 h - 6 x + 4$

Reorder $- 3 h$ and $- 6 x$ .

$- 6 x - 3 h + 4$

Step 5

Find the Difference Quotient f(x)=4x-3x^2+6

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