Solveeit Logo
Login

Question

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

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

Explanation

Solution

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′(x)=lim⁡h→0f(x+h)−f(x)hf'\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 dydx\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:
⇒f′(x)=lim⁡h→0f(x+h)−f(x)h\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:
⇒f′(x)=lim⁡h→0f(x+h)−f(x)h\Rightarrow f'\left( x \right)=\displaystyle \lim_{h \to 0}\dfrac{f\left( x+h \right)-f\left( x \right)}{h}
We will get:
⇒f′(x)=lim⁡h→0m(x+h)+b−(mx+b)h\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:
⇒f′(x)=lim⁡h→0mx+mh+b−mx−bh\Rightarrow f'\left( x \right)=\displaystyle \lim_{h \to 0}\dfrac{mx+mh+b-mx-b}{h}
On cancelling the like terms, we get:
⇒f′(x)=lim⁡h→0mhh\Rightarrow f'\left( x \right)=\displaystyle \lim_{h \to 0}\dfrac{mh}{h}
Cancel ‘h’:
⇒f′(x)=lim⁡h→0(m)\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.
⇒ddxf(x)=ddx(mx+b)\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.