Find the Difference Quotient f(x)=5x^2+2x-1

$f (x) = 5 x^{2} + 2 x - 1$

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.

$f (x + h) = 5 {(x + h)}^{2} + 2 (x + h) - 1$

Simplify the result.

Simplify each term.

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

$f (x + h) = 5 ((x + h) (x + h)) + 2 (x + h) - 1$

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

Apply the distributive property.

$f (x + h) = 5 (x (x + h) + h (x + h)) + 2 (x + h) - 1$

Apply the distributive property.

$f (x + h) = 5 (x \cdot x + x h + h (x + h)) + 2 (x + h) - 1$

Apply the distributive property.

$f (x + h) = 5 (x \cdot x + x h + h x + h \cdot h) + 2 (x + h) - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x + h) = 5 (x^{2} + x h + h x + h \cdot h) + 2 (x + h) - 1$

Multiply $h$ by $h$ .

$f (x + h) = 5 (x^{2} + x h + h x + h^{2}) + 2 (x + h) - 1$

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

$f (x + h) = 5 (x^{2} + h x + h x + h^{2}) + 2 (x + h) - 1$

Add $h x$ and $h x$ .

$f (x + h) = 5 (x^{2} + 2 h x + h^{2}) + 2 (x + h) - 1$

Apply the distributive property.

$f (x + h) = 5 x^{2} + 5 (2 h x) + 5 h^{2} + 2 (x + h) - 1$

Multiply $2$ by $5$ .

$f (x + h) = 5 x^{2} + 10 (h x) + 5 h^{2} + 2 (x + h) - 1$

Apply the distributive property.

$f (x + h) = 5 x^{2} + 10 h x + 5 h^{2} + 2 x + 2 h - 1$

The final answer is $5 x^{2} + 10 h x + 5 h^{2} + 2 x + 2 h - 1$ .

$5 x^{2} + 10 h x + 5 h^{2} + 2 x + 2 h - 1$

Reorder.

Move $2 x$ .

$5 x^{2} + 10 h x + 5 h^{2} + 2 h + 2 x - 1$

Move $5 x^{2}$ .

$10 h x + 5 h^{2} + 5 x^{2} + 2 h + 2 x - 1$

Reorder $10 h x$ and $5 h^{2}$ .

$5 h^{2} + 10 h x + 5 x^{2} + 2 h + 2 x - 1$

Find the components of the definition.

$f (x + h) = 5 h^{2} + 10 h x + 5 x^{2} + 2 h + 2 x - 1$

$f (x) = 5 x^{2} + 2 x - 1$

$f (x + h) = 5 h^{2} + 10 h x + 5 x^{2} + 2 h + 2 x - 1$

$f (x) = 5 x^{2} + 2 x - 1$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{5 h^{2} + 10 h x + 5 x^{2} + 2 h + 2 x - 1 - (5 x^{2} + 2 x - 1)}{h}$

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

$\frac{5 h^{2} + 10 h x + 5 x^{2} + 2 h + 2 x - 1 - (5 x^{2}) - (2 x) - - 1}{h}$

Simplify.

Multiply $5$ by $- 1$ .

$\frac{5 h^{2} + 10 h x + 5 x^{2} + 2 h + 2 x - 1 - 5 x^{2} - (2 x) - - 1}{h}$

Multiply $2$ by $- 1$ .

$\frac{5 h^{2} + 10 h x + 5 x^{2} + 2 h + 2 x - 1 - 5 x^{2} - 2 x - - 1}{h}$

Multiply $- 1$ by $- 1$ .

$\frac{5 h^{2} + 10 h x + 5 x^{2} + 2 h + 2 x - 1 - 5 x^{2} - 2 x + 1}{h}$

Subtract $5 x^{2}$ from $5 x^{2}$ .

$\frac{5 h^{2} + 10 h x + 2 h + 2 x - 1 + 0 - 2 x + 1}{h}$

Add $5 h^{2}$ and $0$ .

$\frac{5 h^{2} + 10 h x + 2 h + 2 x - 1 - 2 x + 1}{h}$

Subtract $2 x$ from $2 x$ .

$\frac{5 h^{2} + 10 h x + 2 h + 0 - 1 + 1}{h}$

Add $5 h^{2}$ and $0$ .

$\frac{5 h^{2} + 10 h x + 2 h - 1 + 1}{h}$

Add $- 1$ and $1$ .

$\frac{5 h^{2} + 10 h x + 2 h + 0}{h}$

Add $5 h^{2} + 10 h x + 2 h$ and $0$ .

$\frac{5 h^{2} + 10 h x + 2 h}{h}$

Factor $h$ out of $5 h^{2} + 10 h x + 2 h$ .

Factor $h$ out of $5 h^{2}$ .

$\frac{h (5 h) + 10 h x + 2 h}{h}$

Factor $h$ out of $10 h x$ .

$\frac{h (5 h) + h (10 x) + 2 h}{h}$

Factor $h$ out of $2 h$ .

$\frac{h (5 h) + h (10 x) + h \cdot 2}{h}$

Factor $h$ out of $h (5 h) + h (10 x)$ .

$\frac{h (5 h + 10 x) + h \cdot 2}{h}$

Factor $h$ out of $h (5 h + 10 x) + h \cdot 2$ .

$\frac{h (5 h + 10 x + 2)}{h}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{h (5 h + 10 x + 2)}{h}$

Divide $5 h + 10 x + 2$ by $1$ .

$5 h + 10 x + 2$

Reorder $5 h$ and $10 x$ .

$10 x + 5 h + 2$

Step 5

Find the Difference Quotient f(x)=5x^2+2x-1

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