What kind of courses are you taking or have taken? If you are asked about derivatives, you should be taking Calculus- but then you should surely know that "f'(ln(1+ cos(x))" is just bad notation. You mean "if f(x)= ln(1+ cos(x)), what is f'(x)?" Now, do you know the chain rule? Clearly, you are expected to because it is needed here. You have f(u)= ln(u) with u= 1+ cos(x). Then . What is the derivative of ln(u) with respect to u? What is the derivative of 1+ cos(x) with respect to x?
Remember, always differentiate from the outside first.
When you differentiate lnx, you'll end up with
So, when you differentiate , you will get
Next, differentiate within the brackets, ie.
Differentiate 1, it will be 0. Differentiate you'll end up with
You have and ,
Thank you very muc-it become clear in just a few minutes!
Please, what if we have derivative of ln(sinx+cosx)?
I supose, we first have 1/(sinx+cosx)
then we have cosx and (-sinx)
so we multiply 1/(sinx+cosx) with (cosx-sinx) and we get (cosx-sinx)/(sinx+cosx)
Am I right?