Question

I am having trouble understanding the solution to a particular problem. The problem is this:

Differentiate f(x)=ln(sin(x)) - (x^4-3x)^10.

I can differentiate the second part of the function, but I dont quite understand differentiating the first part. I want to write (1/x)(sin(x))(cos(x)), but the solution says that it is cos(x)/(sinx(x).

Answer 1

To differentiate the first part, you must use the chain rule. \[d/dx[\ln(\sin(x))]=(1/\sin(x))*\cos(x)=cot(x)\]
Then, you just add the derivative of the first part to the derivative of the second part. :)

Answer 2

Ahh. I thought about that, but thought 1/sin(x) was wrong.

Answer 3

The reason why the derivative of ln(sin(x)) is cos(x)/sin(x) instead of (1/x)*(1/sin(x))*cos(x) is that when you differentiate the natural log, it turns into 1/(whatever what inside) not just 1/x times 1/(whatever was inside)

Answer 4

Great. Thanks for clarifying!

Answer 5

no problem :)