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Problem 1
Find the Derivative of f (x) = 2x+1 using the definition of the derivative:
lim [f(x + Δx) - f(x)] / Δx
Δx-> 0
Solution:
Step1
First replace x with (x-Δx) in the given function f(x) to obtain
f(x+Δx) = 2(x+Δx)+1 = 2x + 2Δx + 1
Step2
Rewrite f(x-Δx) as (2x + 2Δx + 1) and f(x) as (2x + 1) into the definition of the derivative limit to obtain:
lim [(2x + 2Δx + 1) - (2x+1)] / Δx
Δx-> 0
Step 3
Combine like terms on the numerator and the reduced expression becomes
lim [2Δx] / Δx
Δx-> 0
Step 4
Cancel out Δx from both numerator and denominator and finally you get the derivative
Derivative of f (x) = 2x+1 = 2
———-
Note there was no need to substitute Δx with 0 since it was canceled out.