How would I take the partial derivatives of this equation??

Question

Hi, I need the partial derivative of this equation with respect to q. And then I need the partial derivative with respect to p. I am completely lost do I need to use the quotient rule or is there a simplier way?? Can you please show as much working as possible. Thank you so much!!

The equation is:

(2p/q) ^.5 <-- In other words the square root of ((2p)/q).

Thanks so much! :)

Answer 1

Let, y = (2p/q)^1/2

now differentiate wrt q we have,

dy/dq = (1/2).(2p/q)^(1/2 - 1) .2p.(-1/q^2)

=&gt; dy/dq = (1/2).(2p/q)^(-1/2).(- 2p/q^2)

=&gt; dy/dq = (-p/q^2).(2p/q)^(-1/2)

now, y = (2p/q)^1/2

differentite wrt p we have,

dy/dp = (1/2).(2p/q)^(-1/2).(1/q).2

=&gt; dy/dp = (1/q).(2p/q)^(-1/2)

Hope this will help u :)

Answer 2

When doing wrt to q, treat p as constant:

d/dq [(2p)^0.5 * q^(-0.5)]

= (2p)^0.5 * (-0.5) q^(-1.5)

Same concept for wrt to p.