f(x)=((x-4)^(2))/(2x)^(1/2)
a) find the values of the constants A, B and C such that

f(x)= Ax^(3/2)+Bx^(1/2)+C^(-1/2)

b)f'(x)=((3x+4)(x-4))/(4x)^(3/2)

I don't understand how to solve either of these any help would be apprieciated

Work backwards. Integrate f'(x).

How would I do that, bearing in mind this is C1 so it does not cover a lot of the integration

Actually, just expand f(x).

f(x)=\frac{(x-4)^{2}}{\sqrt{2x}}=\frac{1}{\sqrt{2}}x^{\frac{3}{2 }}-4\sqrt{2}\sqrt{x}+\frac{8\sqrt{2}}{\sqrt{x}}

for b) the first step of the answer is
f'x= (3/4)x^(1/2)-2x^(-1/2)-4x^(-3/2)

why is this

Differentiate f(x). That's all. You can cancel out the \sqrt{2}.