What is the derivative of y=(3x^2+2)  cos^2⁡x ?

2020-05-29 12:08 am
My university friend has challenged me to work out the above equation.


In fairness I did give him an equally difficult challenge on sport (as he is non-sporting).


It's just a fun thing that we are doing during lockdown.


Anyway, can somebody give me some help please as I haven't seen this type of equation before?


He says that he can give me another one instead, but I still need to know how to work them out first.


Many thanks.

回答 (3)

2020-05-29 9:48 am
y = ( 3x² + 2 ) cos² ⁡x

Suppose --    ( 3x² + 2 ) =  f1(x)

cos² x  = f2 (x)

then   y =  f1(x) * f 2 (x)

=> dy/dx  =  d f1(x)/dx * f2(x) + d f2(x)/dx * f1(x)  ............. Product Rule

=> dy/dx  = cos² ⁡x * ( 6 x )  +  2 cos x * (-sin x ) * ( 3x² + 2 )

=>  dy/dx = cos² ⁡x * ( 6 x ) -  ( 3x² + 2 ) (sin 2 x )   ..... since 2 sin x * cos x = sin 2x

=> dy/dx = ( 6 x )  cos² ⁡x  - ( 3x² + 2 ) (sin 2 x )  ................. Answer
2020-05-29 3:09 am
y=(3x^2+2) (cos⁡x)^2.  Use product rule.  Recall derivative of cos is -sin.
dy/dx = [ (cos⁡x)^2 ][6x] + [3x^2+2][(2cosx)(-sinx)]
2020-05-29 12:25 am
Sorry. It wouldn't be fair to do your work for you. I'm sure your "Uni friend" would agree.  Ask your friend to work it out in front of you.  S/he shouldn't be asking questions without knowing the correct answer.


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