use the definition find derivative

Question

f(x)=√(5-2x)

 

please explain how to get there

Answer 1

[f(x+h)-f(x)]/h = [√(5-2x-2h) - √(5-2x)]/h

 

Multiply the numerator and denominator by √(5-2x-2h) + √(5-2x) to get

 

[(5-2x-2h)-(5-2x)]/[h(√(5-2x-2h)+√(5-2x)]

 

= -2h/[h(√(5-2x-2h)+√(5-2x)]

 

= -2/[√(5-2x-2h) + √(5-2x)]

 

f'(x) = lim_h→0 ([f(x+h)-f(x)]/h) = lim_h→0[-2/(√(5-2x-2h)+√(5-2x))]

 

                                           = -2/(√(5-2x-0)+√(5-2x))

 

                                           = -2/(√(5-2x)+√(5-2x))

 

                                           = -2/(2√(5-2x))

 

                                           = -1/√(5-2x)

 

 

Answer 2

Hello Karoulin,

 

The possible solution is as following.

Knowing that: (√x)’ = 1/(2√x)

 

f(x)’= -2 * (1/2√(5-2x))  = -1/√(5-2x)

 

Hope this will help.