You are not allowed to ask more than one question. I am answering your first question.

You have given the function as f(x) = sin0.5x+ cos1/3x . I am not sure if the x in the second term is in the denominator or numerator. I take it you mean (1/3)*x

We have the function f(x) = sin (0.5*x) + cos (1/3)*x

The derivative of f(x) can be found using the chain rule.

f'(x) = 0.5*(cos (0.5*x)) - (1/3)*sin (1/3)*x

=> f'(x) = [cos (0.5*x)]/2 - [sin (1/3)*x]/3

**The required derivative of f(x) = sin (0.5)x+ cos (1/3)x is f'(x) = [cos (0.5*x)]/2 - [sin (1/3)*x]/3**