Question

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

Answer 1

`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)`