Find the Difference Quotient f(x)=-x^2+5x+4

$f (x) = - x^{2} + 5 x + 4$

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.

$f (x + h) = - {(x + h)}^{2} + 5 (x + h) + 4$

Simplify the result.

Simplify each term.

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

$f (x + h) = - ((x + h) (x + h)) + 5 (x + h) + 4$

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

Apply the distributive property.

$f (x + h) = - (x (x + h) + h (x + h)) + 5 (x + h) + 4$

Apply the distributive property.

$f (x + h) = - (x \cdot x + x h + h (x + h)) + 5 (x + h) + 4$

Apply the distributive property.

$f (x + h) = - (x \cdot x + x h + h x + h \cdot h) + 5 (x + h) + 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x + h) = - (x^{2} + x h + h x + h \cdot h) + 5 (x + h) + 4$

Multiply $h$ by $h$ .

$f (x + h) = - (x^{2} + x h + h x + h^{2}) + 5 (x + h) + 4$

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

$f (x + h) = - (x^{2} + h x + h x + h^{2}) + 5 (x + h) + 4$

Add $h x$ and $h x$ .

$f (x + h) = - (x^{2} + 2 h x + h^{2}) + 5 (x + h) + 4$

Apply the distributive property.

$f (x + h) = - x^{2} - (2 h x) - h^{2} + 5 (x + h) + 4$

Multiply $2$ by $- 1$ .

$f (x + h) = - x^{2} - 2 (h x) - h^{2} + 5 (x + h) + 4$

Apply the distributive property.

$f (x + h) = - x^{2} - 2 h x - h^{2} + 5 x + 5 h + 4$

The final answer is $- x^{2} - 2 h x - h^{2} + 5 x + 5 h + 4$ .

$- x^{2} - 2 h x - h^{2} + 5 x + 5 h + 4$

Reorder.

Move $5 x$ .

$- x^{2} - 2 h x - h^{2} + 5 h + 5 x + 4$

Move $- x^{2}$ .

$- 2 h x - h^{2} - x^{2} + 5 h + 5 x + 4$

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

$- h^{2} - 2 h x - x^{2} + 5 h + 5 x + 4$

Find the components of the definition.

$f (x + h) = - h^{2} - 2 h x - x^{2} + 5 h + 5 x + 4$

$f (x) = - x^{2} + 5 x + 4$

$f (x + h) = - h^{2} - 2 h x - x^{2} + 5 h + 5 x + 4$

$f (x) = - x^{2} + 5 x + 4$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- h^{2} - 2 h x - x^{2} + 5 h + 5 x + 4 - (- x^{2} + 5 x + 4)}{h}$

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

$\frac{- h^{2} - 2 h x - x^{2} + 5 h + 5 x + 4 - - x^{2} - (5 x) - 1 \cdot 4}{h}$

Simplify.

Multiply $- - x^{2}$ .

Multiply $- 1$ by $- 1$ .

$\frac{- h^{2} - 2 h x - x^{2} + 5 h + 5 x + 4 + 1 x^{2} - (5 x) - 1 \cdot 4}{h}$

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

$\frac{- h^{2} - 2 h x - x^{2} + 5 h + 5 x + 4 + x^{2} - (5 x) - 1 \cdot 4}{h}$

Multiply $5$ by $- 1$ .

$\frac{- h^{2} - 2 h x - x^{2} + 5 h + 5 x + 4 + x^{2} - 5 x - 1 \cdot 4}{h}$

Multiply $- 1$ by $4$ .

$\frac{- h^{2} - 2 h x - x^{2} + 5 h + 5 x + 4 + x^{2} - 5 x - 4}{h}$

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

$\frac{- h^{2} - 2 h x + 5 h + 5 x + 4 + 0 - 5 x - 4}{h}$

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

$\frac{- h^{2} - 2 h x + 5 h + 5 x + 4 - 5 x - 4}{h}$

Subtract $5 x$ from $5 x$ .

$\frac{- h^{2} - 2 h x + 5 h + 0 + 4 - 4}{h}$

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

$\frac{- h^{2} - 2 h x + 5 h + 4 - 4}{h}$

Subtract $4$ from $4$ .

$\frac{- h^{2} - 2 h x + 5 h + 0}{h}$

Add $- h^{2} - 2 h x + 5 h$ and $0$ .

$\frac{- h^{2} - 2 h x + 5 h}{h}$

Factor $h$ out of $- h^{2} - 2 h x + 5 h$ .

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

$\frac{h (- h) - 2 h x + 5 h}{h}$

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

$\frac{h (- h) + h (- 2 x) + 5 h}{h}$

Factor $h$ out of $5 h$ .

$\frac{h (- h) + h (- 2 x) + h \cdot 5}{h}$

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

$\frac{h (- h - 2 x) + h \cdot 5}{h}$

Factor $h$ out of $h (- h - 2 x) + h \cdot 5$ .

$\frac{h (- h - 2 x + 5)}{h}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{h (- h - 2 x + 5)}{h}$

Divide $- h - 2 x + 5$ by $1$ .

$- h - 2 x + 5$

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

$- 2 x - h + 5$

Step 5

Find the Difference Quotient f(x)=-x^2+5x+4

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