Evaluate the Difference Quotient f(x)=4+3x-x^2 , (f(3+h)-f(3))/h

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

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.

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

Simplify the result.

Simplify each term.

Apply the distributive property.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $h$ by $h$ .

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

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

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

Add $h x$ and $h x$ .

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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

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

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

Reorder.

Move $4$ .

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

Move $3 x$ .

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

Move $3 h$ .

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

Move $- x^{2}$ .

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

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

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

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

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

Simplify.

Multiply $- 1$ by $4$ .

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

Multiply $3$ by $- 1$ .

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

Multiply $- - x^{2}$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $x^{2}$ by $1$ .

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

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

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

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

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

Subtract $3 x$ from $3 x$ .

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

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

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

Subtract $4$ from $4$ .

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

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

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

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

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

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

Factor $h$ out of $- 2 h x$ .

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

Factor $h$ out of $3 h$ .

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

Factor $h$ out of $h (- h) + h (- 2 x)$ .

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

Factor $h$ out of $h (- h - 2 x) + h \cdot 3$ .

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

Reduce the expression by cancelling the common factors.

Cancel the common factor of $h$ .

Cancel the common factor.

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

Divide $- h - 2 x + 3$ by $1$ .

$- h - 2 x + 3$

Reorder $- h$ and $- 2 x$ .

$- 2 x - h + 3$

Step 5

Replace the variable $x$ with $3$ in the expression.

$- 2 \cdot 3 - h + 3$

Step 6

Multiply $- 2$ by $3$ .

$- 6 - h + 3$

Step 7

Add $- 6$ and $3$ .

$- h - 3$

Step 8

Evaluate the Difference Quotient f(x)=4+3x-x^2 , (f(3+h)-f(3))/h

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