Having trouble finding the derivative of this function and solving it

[math challenged]

Hi guys, to clarify, this is part of an economics problem where I am supposed to find the maximum value of the total revenue function (what level of q (output) total revenue is at its maximum)
So here is the total revenue function (which I have to get the derivative of and set it to 0 to find the maximum value)
TR = [ ((-1/4)q + 75)^1/2 ] * q

It seems that you have to use the chain and product rules to differentiate this, but I'm 100% sure that halfway into the process I'm making some sort of dumb mistake and everything looks like it goes wrong.

So far, I think I found the derivative of the first term using the chain rule. So [ ((-1/4q) + 75)^1/2 ] has a derivative of (-1/8)[ ((-1/4q) + 75)^-1/2 ]. And q has a derivative of 1. I think the product rule says to multiply each function by the derivative of the other function and sum up. So if I am right, the resulting expression from which to obtain the derivative of the TR function would be:
[ ((-1/4)q + 75)^1/2 ] * (1) + (-1/8)[ ((-1/4q)+75)^-1/2 ] * (q)

Evaluating this is where I seem to go horribLy wrong. Or maybe I made a mistake before even reaching this stage, though I don't think I did. Sorry for the long post, any help would be greatly appreciated, thank you!
Just a note: this is not part of a class, just learning this kind of stuff and the calculus that comes with it at my own leisure.

 

[Steven G]

I did not see any error in the derivative work. Good job!

Now get common denominators and see where the function is 0 or undefined.

Post back

 

[Deleted member 4993]

Hi guys, to clarify, this is part of an economics problem where I am supposed to find the maximum value of the total revenue function (what level of q (output) total revenue is at its maximum)
So here is the total revenue function (which I have to get the derivative of and set it to 0 to find the maximum value)
TR = [ ((-1/4)q + 75)^1/2 ] * q

It seems that you have to use the chain and product rules to differentiate this, but I'm 100% sure that halfway into the process I'm making some sort of dumb mistake and everything looks like it goes wrong.

So far, I think I found the derivative of the first term using the chain rule. So [ ((-1/4q) + 75)^1/2 ] has a derivative of (-1/8)[ ((-1/4q) + 75)^-1/2 ]. And q has a derivative of 1. I think the product rule says to multiply each function by the derivative of the other function and sum up. So if I am right, the resulting expression from which to obtain the derivative of the TR function would be:
[ ((-1/4)q + 75)^1/2 ] * (1) + (-1/8)[ ((-1/4q)+75)^-1/2 ] * (q)

Evaluating this is where I seem to go horribLy wrong. Or maybe I made a mistake before even reaching this stage, though I don't think I did. Sorry for the long post, any help would be greatly appreciated, thank you!
Just a note: this is not part of a class, just learning this kind of stuff and the calculus that comes with it at my own leisure.

If I were to evaluate this "horrible" function for any given value of 'q' - i would break it up in the following way

[ ((-1/4)q + 75)^1/2 ] * (1) + (-1/8)[ ((-1/4q)+75)^-1/2 ] * (q)

I will define a mini-function:

A = ((-1/4)q + 75)

Then we have:

[ (A^1/2 ] * (1) + (-1/8)[ (A^-1/2 ] * (q)

= \(\displaystyle \frac{A- \frac{q}{8}}{\sqrt{A}}\)

=\(\displaystyle \frac{75 - \frac{3q}{8}}{\sqrt{A}}\)

Now apply the conditions for the function to exist and real (A > 0)

and apply the condition for root of the function