QUESTION: Greetings,

I look at calculus books and examples on various internet sites and I don't see the "hops" in manipulation of the terms.
I seem to be getting tripped up on problems at the point of recognizing where to parse the expression to apply differential rules.  Do you have some method of identifying what to manipulate a given expression to in order to solve it? Can you elaborate on this aspect of "figuring out what to do" when presented a problem.  e.g. find the second derivative of y= 3x cos (x^2).  Thanks!

ANSWER: The problem is a product rule.
We have y = f(x)g(x), so y' = f'(x)g(x) + f(x)g'(x).

Since f(x) = 3x, f'(x) = 3.
Since g(x) = cos(3x), the chain rule is used to get g'(x) = -sin(3x)[6x].
Putting the 6x in front gives -6x*sin(3x).

Putting these in gives y' = 3cos(3x) - 18xsin(3x).

To find y", the 1st term is done with the chain rule alone and
the 2nd term is done with the product rule and chain rule.

For 3*cos(3x), the derivative of cos() is -sin() and the derivative of 3x is 6x.
Putting the minus sign and multiplying the front by 6 gives This gives -18x*sin(3x).

For -18xsin(3x), the product rule says to take f(x) = - 18x and g(x) = sin(3x).
This gives f'(x) = -36x and g'(x) = 6x*cos(3x).  That makes the derivative fg' + gf' be
[-18x][6x*cos(3x)] +  sin(3x)[-36x].

Multiplying terms together gives -108x*cos(3x)] - 36x*sin(3x).
Note that we could factor out -36x, giving -36x[-3x*cos(3x)] + sin(3x)].

---------- FOLLOW-UP ----------

QUESTION: Thanks for given my question a try Scotto. Given: y= 3x cos (x^2)

F(x) = 3x derivative becomes f'(x)= 3

G(x) is cos x^2 so (cos x^2) becomes - sin x^2 and derivative of x^2 becomes (2x) right?

so I got lost on why you have it =  -sin 3x^2 (6x)

I never got to the rest because I got lost here, the rest will probably make perfect sense once I get over this stumbling block.

For some reason I put a 3 in the cos(), and as you pointed out, its not there.
We have y = 3x cos(x).
This makes y' = 3x * (-2x sin(x) + 3 cos(x).
That is, y' = -6x sin(x) + 3 cos(x).
This makes y" = -6x * 2x*cos(x) - 6 sin(x) - 3*2x*sin(x).
That works out to y" = -12x*cos(x) - 6(1+x)sin(x).


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