How can one derive the formula for the derivative of sec(x), csc(x) and cot(x) with respect to x? What exactly should be done here?
You have to derive the derivative for the three trigonometric formulas which are given.
If this question is right at the beginning of your course, just after you have learnt to find the derivative from first principles, you would have to find each of them using:
For the three questions f(x) = sec x , f(x) = csc x and f(x) = cot x resp.
If you have already learnt the rules like quotient rule and product rule to find derivatives of complex functions, you can use those.
The quotient rules gives the derivative of f(x)=g(x)/h(x) as f'(x)=[g'(x)*h(x) - g(x)*h'(x)]/(h(x))^2
sec x = 1/cos x; use the quotient rule here to find the derivative of sec x with the derivative of cos x taken as -sin x.
Similarly, the derivative of sin x is cos x, use that and the fact that csc x = 1/sin x and finally use the quotient rule and the derivatives of sin x and cos x to find the derivative of cot x as cot x = cos x/sin x.