What is the derivative of f(x) = xcos^3(x^2)+sin(x)?

1 Answer
Apr 28, 2017

cos^3(x^2) - 6x^2cos^2(x^2)sin(x^2) + cos(x)

Explanation:

Firstly we use the product rule, (xcos^3(x^2))' = x'cos^3(x^2)+x(cos^3(x^2))'

Now we must use the chain rule on the second term.
(cos^3(x^2))' = 3(cos^2(x^2))(-sin(x^2))(2x))
:. = (1)cos^3(x^2) + x*3(cos^2(x^2))(-sin(x^2))(2x))
:. = cos^3(x^2) - 6x^2cos^2(x^2)sin(x^2)
lastly the derivative of sin(x) = cos(x)

Thus, the complete derivative is cos^3(x^2) - 6x^2cos^2(x^2)sin(x^2) + cos(x)