# Inverse Logarithmic Differentiation

Hey everyone,
I'm having some trouble with a couple HW questions. A little help would be greatly appreciated.
1. Find the derivative of:
y=2(arcsin(x^2))
I thought you had to make a u-substition of u=x^2, but it doesnt seem to work out. If anyone could point me in the right direction that would be good.
2. Find the derivative of: y=ln(x^7(x+5)^9((x^2)+9)^2)
For this I took the ln of both sides, then tried using the chain rule over and over, but again i didnt get the answer.
If anyone could guide me in the right direction I would be really
Thanks

StatusX
Homework Helper
For the first one, u substitution is for integrals. It is the integral equivalent of the chain rule, which is what you should be doing here. Do you know the derivative of arcsin(x)?

For the second, I don't know why you'd take the log of both sides, leaving a nested log on the right side, which only makes things harder. Just expand the log into a sum of the logs of the factors and then appy the chain rule to each term.

By the way, I should point out this thread has a very misleading title. One function is an inverse trig and one is a log, and your differentiating each of them. You aren't differentiating inverse logs (exponents), nor finding the antiderivatives of logs. Just saying.

Last edited:
dextercioby
Homework Helper

$$\ln\left(x^{7(x+5)^{9(x^{2}+9)^{2}}}\right)$$

and u need to differentiate wrt to "x". It's not difficult and indeed you need to use logarithmic differentiation when you have to differentiate the exponent of "x".

Daniel.

HallsofIvy
Homework Helper
dextercioby said:
$$\ln\left(x^{7(x+5)^{9(x^{2}+9)^{2}}}\right)$$
and u need to differentiate wrt to "x". It's not difficult and indeed you need to use logarithmic differentiation when you have to differentiate the exponent of "x".
Daniel.

No, I believe he said " y=ln(x^7(x+5)^9((x^2)+9)^2)" which, copied directly into TEX, is
$$y=ln(x^7(x+5)^9((x^2)+9)^2)$$

That is, of course, the same as
y= 7ln(x)+ 9ln(x+5)+ 2ln(x2+ 9).