The question "what is the derivative of theta?" is not clear as it does not specify the function with respect to which theta is being differentiated. Theta is an angle and not a function, so it cannot be differentiated on its own. However, if theta is part of a function, then its derivative can be found using the chain rule and the general power rule of differentiation. For example, if we have a function f(theta) = 3sin^2(pi - theta), then its derivative with respect to theta can be found as follows:
- Apply the chain rule: df/dtheta = df/du * du/dtheta, where u = pi - theta.
- Find the derivative of u with respect to theta: du/dtheta = -1.
- Find the derivative of f with respect to u: df/du = 6sin(pi - theta)cos(pi - theta).
- Substitute u back into the expression: df/dtheta = 6sin(pi - theta)cos(pi - theta) * (-1) = -6sin(pi - theta)cos(pi - theta).
Therefore, the derivative of f(theta) = 3sin^2(pi - theta) with respect to theta is -6sin(pi - theta)cos(pi - theta) .
