Question

How do you find the derivative of the function f(x) = mx + b?

Answer 1

Hint: In this particular problem. We are given a linear equation in one variable which consists of a slope and an intercept. We have to find the first order derivative of the given equation. The definition of derivative is:
$f'\left( x \right)=\displaystyle \lim_{h \to 0}\dfrac{f\left( x+h \right)-f\left( x \right)}{h}$

Complete step-by-step answer:
Now, let’s discuss the question.
As we all know that Differentiation can be defined as a derivative of a function with respect to an independent variable. It can be applied to measure the function per unit change in the independent variable. Let y = f(x) which is a function of x. Then, the rate of change of “y” per unit change in “x” is given by $\dfrac{dy}{dx}$. But if the function f(x) undergoes an infinite change of ‘h’ near to any point ‘x’, then the derivative of the function can be defined as:
$\Rightarrow f'\left( x \right)=\displaystyle \lim_{h \to 0}\dfrac{f\left( x+h \right)-f\left( x \right)}{h}$
So we can also say that the rate of change of a function at a particular point is defined as a derivative of that particular function.
We are given:
$\Rightarrow $f(x) = mx + b
On applying the formula to find the derivative of the function:
$\Rightarrow f'\left( x \right)=\displaystyle \lim_{h \to 0}\dfrac{f\left( x+h \right)-f\left( x \right)}{h}$
We will get:
$\Rightarrow f'\left( x \right)=\displaystyle \lim_{h \to 0}\dfrac{m\left( x+h \right)+b-\left( mx+b \right)}{h}$
Now, on opening brackets, we will get:
$\Rightarrow f'\left( x \right)=\displaystyle \lim_{h \to 0}\dfrac{mx+mh+b-mx-b}{h}$
On cancelling the like terms, we get:
$\Rightarrow f'\left( x \right)=\displaystyle \lim_{h \to 0}\dfrac{mh}{h}$
Cancel ‘h’:
$\Rightarrow f'\left( x \right)=\displaystyle \lim_{h \to 0}\left( m \right)$
So we get:
$\therefore $f’(x) = (m)
This is the final answer.

Note: There is a simple shortcut method also for this question. We need to find the first order derivative by applying differentiation directly.
$\Rightarrow \dfrac{d}{dx}f(x)=\dfrac{d}{dx}\left( mx+b \right)$
So, derivative will be:
$\Rightarrow $f’(x) = m
So we got the same answer by both the methods. Differentiation of a constant is zero.