Find the Difference Quotient f(x)=10x+5

$f (x) = 10 x + 5$

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.

$f (x + h) = 10 (x + h) + 5$

Simplify the result.

Apply the distributive property.

$f (x + h) = 10 x + 10 h + 5$

The final answer is $10 x + 10 h + 5$ .

$10 x + 10 h + 5$

Reorder $10 x$ and $10 h$ .

$10 h + 10 x + 5$

Find the components of the definition.

$f (x + h) = 10 h + 10 x + 5$

$f (x) = 10 x + 5$

$f (x + h) = 10 h + 10 x + 5$

$f (x) = 10 x + 5$

Step 3

Plug in the components.

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

Step 4

Simplify.

Simplify the numerator.

Factor $5$ out of $10 x + 5$ .

Factor $5$ out of $10 x$ .

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

Factor $5$ out of $5$ .

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

Factor $5$ out of $5 (2 x) + 5 (1)$ .

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

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

Multiply $- 1$ by $5$ .

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

Factor $5$ out of $10 h + 10 x + 5 - 5 (2 x + 1)$ .

Factor $5$ out of $10 h$ .

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

Factor $5$ out of $10 x$ .

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

Factor $5$ out of $5$ .

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

Factor $5$ out of $- 5 (2 x + 1)$ .

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

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

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

Factor $5$ out of $5 (2 h + 2 x) + 5 (1)$ .

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

Factor $5$ out of $5 (2 h + 2 x + 1) + 5 (- (2 x + 1))$ .

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $1$ .

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

Subtract $2 x$ from $2 x$ .

$\frac{5 (2 h + 0 + 1 - 1)}{h}$

Add $2 h$ and $0$ .

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

Subtract $1$ from $1$ .

$\frac{5 (2 h + 0)}{h}$

Add $2 h$ and $0$ .

$\frac{5 \cdot 2 h}{h}$

Multiply $5$ by $2$ .

$\frac{10 h}{h}$

Divide $10$ by $1$ .

$10$

Step 5

Find the Difference Quotient f(x)=10x+5

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