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.