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).

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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).

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

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Great. Thanks for clarifying!
no problem :)

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