Derivative Help

[courtrigrad]

Hello all:

I need help in finding the derivative of:

log( sqrt(1+ log x) - sin x )

I know that derivative of log x is 1/x.

1/ sqrt(1 + logx) - sin x ) = sqrt(1+log x) + sin x / ( 1 + log x - sin ^2 x)

Then I found derivative of inside expression and multiplied with the previous derivative. I get something almost the same as the answer, but I can't seem to simplify it.

The answer is:

(1- 2x)* sqrt(1 + log x cos x) / 2x * sqrt(1 + log x)* sqrt(1 + log x - sin x)

Any help is greatly appreciated

Thanks!

[mprm86]

Using the chain rule, if you make u = (sqrt(1+logx) - sinx), then the derivative will be
1/u * du/dx, where du/dx is:
1/2sqrt(1+logx) * 1/x - cosx

I hope this will help. When you are derivating sqrt(1+logx), you have to use again the cahin rule, so it will be sqrt(v), and the derivative will be:
1/2sqrt(v) * v´