# The Chain Rule

Published on: Fri Dec 05 2008

Practicing problems from UCDavis.
Problem 1
Problem 2
Note to Self: Square roots. To find the derivative of a square root, change the square root into a fraction and proceed, using the exponent rule. Ie. The derivative of the square root of x is 1/2x^{-1/2}
Problem 3
According to the shown solution this answer is wrong. I should have included (-4+35x^{4}) inside of the exponent “29”. I don’t know why I should do this. Is it because a simple version of this might look like 3(x^{5})^{2}(x^{4})? I believe with that sort of problem I could write out 3(x^{5})(x^{4})^{2} and if I solve either one of those two I would get the same answer.
Lets try that:
Let x = 2
3(x^{5})^{2}(x^{4})
3(32)^{2}(14)
3(1024)(14)
43008
3(2^{5})(2^{4})^{2}
3(32)(14)^{2}
1806336
That’s not right either.
I’m going to have to do some research on this. And in the mean time make a mental note, to move the values bracketing the (u) formula to the end! *Solved; See Problem 6*
Problem #4
OK… So I answered this one wrong as well. (The correct answer is shown though..) This time I should not have pulled g’(x) inside of the exponent, like I needed to in problem #3. So what is going on here? How to do I handle exponents? *Solved; See Problem 6*
As a side note. I’ve learned how to “Align along = sign” How nice!
Problem #5
Ok. This one immediately threw me for a loop with the division AND the exponent. First things first, I was able to reduce *u*, leaving the inner function division free but with negative exponents. After that it was smooth flowing, except by the end of my problem, I still have three pesky negative exponents which I need to simplify out of existence.
Problem #6
I got this one right! And it was EASY!
D_{x}y Sin(x) = Cos(x)
I believe I’ve also solved the exponent trouble I was having. The trick is to simplify first when *u* is still a variable, and then once simplified completely, replace *u*