What is the derivative of this?

Question

I need the derivative of this so I can plug in 25.

30 ((70-p)/p)^1/2

I have to use so many different rules that it's confusing me and making me want to cry. Please help me. Thanks.

Answer 1

First I'd use the Chain Rule which is:
(f o h)(x) = f'(h(x)) * h'(x)

f(x) = (h(x)) ^ 1/2
f'(x) = 1/2 * (h(x))^ (-1/2)

h(x) = (70-p)/p
To find h'(x), you have to use the Quotient Rule which is:
(f/g)'(x) = (g(x) f'(x) - f(x)g'(x))/ (g(x)) ^2
g(x) = p ; f(x) = 70 - p ; g'(x) = 1 ; f'(x) = -1
Plug it in you get:
h'(x) = (p * (-1) - (70-p)* 1/ p^2 = (-p - 70 + p)/p ^2 = -70/p^2
h'(x) = -70/ p^2

Now to apply the chain rule

Dx = 30 * 1/2* (h(x))^ (-1/2) * h'(x)
Dx = 15 * ((70-p)/p) ^ (-1/2) * -70/ p^2
Dx = -1050/p^2 * ((70-p)/p) ^ (-1/2)

Hope that helps! =)