derivative of

f(x)=arctan(x^2-x)
is it the product rule or the chain rule?

Chain rule. First the derivative of arctan, then derivative of (x^2=x)

To find the derivative of the function f(x) = arctan(x^2 - x), we need to use the chain rule. The chain rule is used when we have a composition of functions, such as applying the arctan function to the quantity (x^2 - x).

Here's how to apply the chain rule:

1. Identify the inner function, which is (x^2 - x).
2. Find the derivative of the inner function with respect to x.
- Applying the power rule, the derivative of x^2 is 2x.
- Applying the power rule, the derivative of x is 1.
- So, the derivative of (x^2 - x) is (2x - 1).
3. Identify the outer function, which is arctan(u), where u = (x^2 - x).
4. Find the derivative of the outer function with respect to u.
- The derivative of arctan(u) is 1 / (1 + u^2).
5. Apply the chain rule by multiplying the derivatives from steps 2 and 4.
- So, the derivative of f(x) = arctan(x^2 - x) is
(1 / (1 + (x^2 - x)^2)) * (2x - 1).

Therefore, the derivative of f(x) = arctan(x^2 - x) is given by
f'(x) = (2x - 1) / (1 + (x^2 - x)^2).