Find the derivative (chain rule)?

Question

Please show and explain all steps. The best answer gets the points! 

f(x) = x^2(x-2)^4 

f(x) = x*(sqrt(1-x^2)) 

f(x) = ((x+5)/(x^2+2))^2

Answer 1

(1)

Chain Rule:

Chain rule is defined as a mathametical rule concerning the differentiation of a function of a function (as f[u(x )] ) by which under suitable conditions of continuity and differentiability, one function is differentiated with respect to the second function considered as an independent variable and then the second function is differentiated with respect to its independent variable.

Chain rule allows functions that are composed of one another to be differentiated.

If y = f (u ) and u = g (x ) then the derivative of y is , Leibnitz notation.

The function is

Differentiate with respect to x.

Apply Product rule: 

Answer 2

(3)

The function is 

Differentiate with respect to x.

 

Answer 3

(2)

The function is 

Differentiate with respect to x.