y=(x^5-3x^3)cos(2+3x-5x^7) --- Derivative?

Question

A little confused on this one.

Answer 1

It's a product:

if f(x) = a*b
then f'(x) = a*(b') + b*(a')

Also, you must multiply by the derivative of "what's inside." For this problem, you must take the derivative of "what's inside" the cosine parentheses (2 + 3x - 5x^7).

So for this one, a = x^5 - 3x^3 and b = cos(2 + 3x - 5x^7). "What's inside" is (2 + 3x - 5x^7). Therefore:

--- --- --- --- --- --- --- --- --- ---- ---- --- "what's inside"
f'(x) = ( (x^5 - 3x^3)(-sin(2 + 3x - 5x^7)*(3 - 35x^6)) + (cos(2 + 3x - 5x^7)*(5x^4 - 9x^2))

Answer 2

Use Product Rule. The derivative of a product is the first times the derivative of the second plus the derivative of the first times the second.

dy/dx= (x^5 -3x^3)(-sin(2+3x-5x^7))(3+35x^6)+
cos(2 +3x-5x^7)(5x^4 -9x^2)

I'll leave it up to you to attempt to simplify this result. Leaving it "as is" is just fine.