Chain Rule Help

1. Mar 3, 2005

dec1ble

I am presenting a problem in front of the class tomorrow and I am slightly confused on the steps for my problem. The problem is:

Find the derivative of the given function

f(w) = ln[cos(w-1)]

The answer in the back of my book shows the derivative is -tan(w-1) - but I'm my steps aren't giving that answer - could anyone show me the steps to use in order to get that answer? Thanks a lot!

2. Mar 3, 2005

dextercioby

What's the cain rule for $$\frac{d}{dx} \ln u(x)$$...?

Daniel.

3. Mar 3, 2005

dec1ble

well the derivative of (ln x) is 1/x

4. Mar 3, 2005

dextercioby

I didn't ask u that...Oh,DO NOT DOUBLE POST...!!!

I meant for a logarithm whose argument is a general function u(x),not the particular value u(x)=x...

Daniel.

5. Mar 3, 2005

hint: $$\frac{d}{dx} \ln u = \frac{1}{u} du$$ Now what is u?

Last edited: Mar 3, 2005
6. Mar 3, 2005

dec1ble

see thats where im confused.....i believe i have to use the chain rule twice....but im just confused on where to start and how to integrate in the ln

7. Mar 3, 2005

dec1ble

the u would be the cos(w-1) right ?

8. Mar 3, 2005

yes. now you have to find the dervative of u and multiply it by your previous result.

9. Mar 3, 2005

dextercioby

Daniel.

10. Mar 3, 2005

dec1ble

alright - so when i find the derivative of cos(w-1) - it will be -sin(1) - making the equation -sin(1)/cos(w-1), making -sin/cos = -tan(w-1) the answer

thanks for all your guys help - much appreciated

11. Mar 3, 2005

Jameson

To clarify, $$\frac{d}{dx}\ln{u} = \frac{1}{u}du$$.

The "du" is very important...

Daniel... what formula were you referring to as incorrect?

12. Mar 3, 2005

dextercioby

There are 2 now:the first and the last.The correct one is:
$$\frac{d(\ln u)}{dx}=\frac{1}{u}\frac{du}{dx}$$

Daniel.

13. Mar 3, 2005

Jameson

Ah, I see now. Good point.

14. Mar 3, 2005

SomeRandomGuy

The derivative of cos(w-1) is not equal to -sin(1). An easy way for me to understand the chain rule is to say to myself "derivative of the outside times the derivative of the inside". What's the derivative of cos(x)? Now, when you have that, sub in x as w-1, then find the derivative of w-1.

15. Mar 4, 2005

BobG

You're on the right track thinking you need to apply the chain rule twice, but you applied it wrong. The derivative of cos(u) is -sin(u)*du

In this case, the u=(w-1), and the derivative of (w-1) is 1. You should have -sin(w-1)*1. Now you wind up with:

$$\frac{-sin(w-1)}{cos(w-1)} = -tan(w-1)$$