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

1 Answer
kumail
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)#

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