OpenStudy (anonymous):

This one should be simple: Find y' and y'' of y = cos (x^2) I believe the first derivative should be: y' = (cos x^2)' = (cos (x^2)' + cos' x^2 = cos 2x - sin x^2 Where is my error? I cannot find the second derivative if the first one is in correct! Thank you!

4 years ago
OpenStudy (anonymous):

Do I need to further derive the second term: -sin x^2?

4 years ago
OpenStudy (amistre64):

dont confuse the name of a function as tho it is a variable: f(x) is NOT f * x cos is the name of a function; call it f f(x^2) derives to f'(x^2) * (x^2)' it is not a product rule

4 years ago
OpenStudy (amistre64):

y' = (cos x^2)' = (cos (x^2)' * (x^2)' = -sin (2x) * 2x

4 years ago
OpenStudy (amistre64):

y' is now the product of 2 things, to work into y''

4 years ago
OpenStudy (anonymous):

I am trying to understand. I understood that cos was the function being applied to the term x^2. And to derive it I had to derive by applying the product rule: cos x^2 = (cos)' x^2 + cos (x^2)' You are saying that is incorrect. I simply read the problem as: (cos x^2)' = cos x^2 * (x^2)'? So the final is cos x^2 * 2x?

4 years ago
OpenStudy (amistre64):

"And to derive it I had to derive by applying the product rule"; that is not correct the product rule is for when you have a multiplication of 2 or more terms: a*b is a product of "a" and "b" f(x) is NOT a product of "f" and "x"

4 years ago
OpenStudy (anonymous):

Argh! So I DO derive the cosine as well. I am confusing when to leave a term like "cos" alone and when to derive it! (cos x^2)' = (cos)' * x^2 + cos * (x^2)' = - sin x^2 + cos 2x (and this must be further derived) = - sin x^2 + (cos 2x)' = - sin x^2 + (cos)' 2x + cos (2x)' ?

4 years ago
OpenStudy (amistre64):

the "popout" rule ... to name it better, applies when a function wraps around anouther function: |dw:1373463943348:dw|

4 years ago