#### To determine

**To find:** The derivative of the function f(x)=mx+b and state the domain of the function and its derivative.

#### Answer

The derivative of the function f(x) is *m*.

The domain of the function f(x) is ℝ

The domain of f′(x) is ℝ.

#### Explanation

**Formula used:**

The derivative of a function *f ,* denoted by f′(x), is

f′(x)=limh→0f(x+h)−f(x)h (1)

**Calculation:**

Obtain the derivative of the function f(x).

Compute f′(x) by using the equation (1).

f′(x)=limh→0f(x+h)−f(x)h=limh→0(m(x+h)+b)−(mx+b)h=limh→0mx+mh+b−mx−bh=limh→0mhh

Since the limit *h* approaches zero but not equal to zero, cancel the common term *h* from both the numerator and the denominator,

f′(x)=limh→0m=m

Thus, the derivative of the function f(x) is *m*.

The domain of the function f(x) is ℝ since every x∈ℝ there is unique image f(x)∈ℝ.

The domain of the derivative f′(x) is {x∈ℝ| f′(x) exists}.

Since the derivative f′(x)=m exists for all real numbers.

Therefore, the domain of f′(x) is ℝ.