Use the Limit Definition of the Derivative to solve the following:

Question

𝑓(𝑥) = √(𝑥 + 2) , find 𝑓 ′ (2)

Answer 1

f'(x) = Lim_h-->0 { ( f(x+h) - f(x) ) / h }

= Lim_h-->0 { ( (x+h+2)^1/2 - (x+2)^1/2 ) / h }

Multiply by ((x+h+2)^1/2+(x+2)^1/2 ) / ((x+h+2)^1/2+(x+2)^1/2 ) (notice this is 1)

and using (a-b)(a+b) = a²-b² :

= Lim_h-->0 { ( (x+h+2) - (x+2) ) / (h * (x+h+2)^1/2+(x+2)^1/2 ) }

= Lim_h-->0 { 1 / ( (x+h+2)^1/2+(x+2)^1/2 ) }

= 1 / (2*(x+2)^1/2)

f' (2) = 1/4