SOLUTION: Find the derivative of the function: h(x) = ln(cos(x^2)) Please explain how to solve Thanks

Algebra ->  Test -> SOLUTION: Find the derivative of the function: h(x) = ln(cos(x^2)) Please explain how to solve Thanks      Log On


   

Question 983092: Find the derivative of the function:
h(x) = ln(cos(x^2))
Please explain how to solve
Thanks
Found 2 solutions by Fombitz, Alan3354:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Multiple application of the chain rule.
dh%2Fdx=%28dh%2Fdv%29%28dv%2Fdu%29%28du%2Fdx%29
.
.
h=ln%28v%29
dh%2Fdv=1%2Fv
.
.
v=cos%28u%29
dv%2Fdu=-sin%28u%29
.
.
u=x%5E2
du%2Fdx=2x
So then,
dh%2Fdx=%281%2Fcos%28x%5E2%29%29%2A%28-sin%28x%5E2%29%2A2x%29
dh%2Fdx=-2x%2Atan%28x%5E2%29

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
h(x) = ln(cos(x^2))
-------------
It's the "chain rule."
--> 1/(cos(x^2)) 1st term
times -sin(x^2) 2nd term
times 2x 3rd term
-------------------
= -2x%2Asin%28x%5E2%29%2F%28cos%28x%5E2%29%29
= -2x%2Atan%28x%5E2%29