Question

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

Answer 1

Hint: Differentiate both the sides with respect to the variable x. Use the formula: $\dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}}$, where n must not be 0, to find the derivative of the term ax. Use the fact that the derivative of a constant term is 0 to get the answer.

Complete step by step solution:
Here, we have been provided with the function f (x) = ax + b and we are asked to find its derivative, that means, we have to differentiate the function f (x) to find f’(x).
$\because f\left( x \right)=ax+b$
Clearly, we can see that the f (x) is a function of variable x, so on differentiating both the sides with respect to x, we get,
$\Rightarrow \dfrac{d\left[ f\left( x \right) \right]}{dx}=\dfrac{d\left( ax+b \right)}{dx}$
Breaking the terms we get,
$\Rightarrow \dfrac{d\left[ f\left( x \right) \right]}{dx}=\dfrac{d\left( ax \right)}{dx}+\dfrac{d\left( b \right)}{dx}$
Here, we can see that the coefficient of x is ‘a’ which is a constant, so it can be taken out of the derivative, so we have,
$\begin{align}
  & \Rightarrow \dfrac{d\left[ f\left( x \right) \right]}{dx}=a\dfrac{d\left( x \right)}{dx}+\dfrac{d\left( b \right)}{dx} \\
 & \Rightarrow \dfrac{d\left[ f\left( x \right) \right]}{dx}=a\dfrac{d\left( {{x}^{1}} \right)}{dx}+\dfrac{d\left( b \right)}{dx} \\
\end{align}$
Now, using the formula: $\dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}}$, where n =1 in the above relation, we get,
$\begin{align}
  & \Rightarrow \dfrac{d\left[ f\left( x \right) \right]}{dx}=a\left( 1\times {{x}^{1-1}} \right)+\dfrac{d\left( b \right)}{dx} \\
 & \Rightarrow \dfrac{d\left[ f\left( x \right) \right]}{dx}=a\left( {{x}^{0}} \right)+\dfrac{d\left( b \right)}{dx} \\
 & \Rightarrow \dfrac{d\left[ f\left( x \right) \right]}{dx}=a\left( 1 \right)+\dfrac{d\left( b \right)}{dx} \\
 & \Rightarrow \dfrac{d\left[ f\left( x \right) \right]}{dx}=a+\dfrac{d\left( b \right)}{dx} \\
\end{align}$
Let us come to the constant term b. We know that the derivative of a constant term is 0, so we have,
$\begin{align}
  & \Rightarrow \dfrac{d\left[ f\left( x \right) \right]}{dx}=a+0 \\
 & \Rightarrow \dfrac{d\left[ f\left( x \right) \right]}{dx}=a \\
\end{align}$

Hence, the derivative of the given function is ‘a’ which is our answer.

Note: Remember that in the formula: $\dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}}$ ‘n’ must not be equal to 0. If that happens then the value of the function ${{x}^{n}}$ will become equal to 1, which is a constant, and its derivative will become 0. The above formula that we have applied to solve the question is known as the power reduction formula in derivatives. You must remember all the basic rules and formulas of differentiation like: - the product rule, chain rule, \[\dfrac{u}{v}\] rule etc. as they are frequently used in both differential and integral calculus.