Find the derivative function for the following functions using the first principles definition of the derivative limh→0f(x+h)−f(x)h . Show all your steps and include proper notation.

Question

a. (4) ƒ(x) = 5x3
b. (5) ƒ(x) = 12x
c. (6) ƒ(x) = √2x

Answer 1

I'll do the 3rd one. The other two are easy.
(f(x+h) - f(x))/h = (√(2(x+h)) - √(2x))/h
= (√(2(x+h)) - √(2x)) (√(2(x+h)) + √(2x)) / (h (√(2(x+h)) + √(2x)))
= (2(x+h) - 2x) / (h (√(2(x+h)) + √(2x)))
= 2h / (h (√(2(x+h)) + √(2x)))
= 2/((√(2(x+h)) + √(2x)))
Now take the limit as h→0 and you get
2/(2√(2x)) = 1/√(2x)

For polynomials, it is a lot more direct.