Chain Rule of Differentiation?

Question

Given y = f(x) with f(1) = 4 and f'(1) = 3, find
a. g'(1) if g (x) = squareroot of f(x)
b. h'(1) if h(x) = f(squareroot of x)

Answer 1

since g(x)=root(f(x)

g'(x)=[1/2*root(f(x))]*f'(x)

substituting 1, 3/4

similarly do the other..

Answer 2

g (x) = squareroot of f(x)
g' (x) = 1/2(squareroot of f(x))^ -1* f"(x)

g' (1) = 1/2(squareroot of f(1))^ -1* (3)
g' (1) = 1/2(sqrt 4))^ -1* (3)
g' (1) = (1/2)(2))^ -1* (3)
g' (1) = (1/2)(1/2))* (3)=3/4.............i
h(x) = f(squareroot of x)
h'(x) = f'(sqrt 0f x)(1/2)(1/(sqrt of x)
h'(1) = f'(1)(1/2)(1/(1)
h'(1) = 3(1/2)(1/(1) =3/2.................ii

Answer 3

a)
g(x)=sqrtf(x)
g'(x)=1/2*f(x)^-1/2*f'(x)
=(1/2)f'(x)/sqrtf(x)
Therefore
g'(1)=(1/2)f'(1)/sqrt f(1)
=1/2*3/sqrt4
=1/2*3/2=3/4
b)
h(x)=f(sqrtx)
h'(x)=f'(sqrtx)*1/2*x^-1/2
=1/2*f'(sqrtx)/x^1/2
h(1)=1/2*f'(sqrtx)/1
=f'(sqrtx)/2[value of f'(sqrtx) not provided]

Answer 4

Elementary...............