Question

find the derivative with respect X of f(x)=ax+b
(by the definition of the derivative)

Answer 1

Explanation:

Here we need to find the derivative of:

$f(x)=ax+b$

with respect to x. In order to do so, we need to use the Definition of the derivative that tells us:

$f'(x)=\underset{\Delta x\rightarrow0}{lim}\frac{f(x+\Delta x)-f(x)}{\Delta x}$

So:

$f'(x)=\underset{\Delta x\rightarrow0}{lim}\frac{a(x+\Delta x)+b-(ax+b)}{\Delta x} \\ \\ f'(x)=\underset{\Delta x\rightarrow0}{lim}\frac{ax+a\Delta x+b-ax-b}{\Delta x} \\ \\ f'(x)=\underset{\Delta x\rightarrow0}{lim}\frac{ax-ax+a\Delta x+b-b}{\Delta x} \\ \\ f'(x)=\underset{\Delta x\rightarrow0}{lim}\frac{a\Delta x}{\Delta x} \\ \\ \boxed{f'(x)=a}$