How would i differentiate the following example in terms of t (x and y are functions of t)

4 sinx cosy = 1


Well, if x and y were complicated functions of t, for instance x(t) = t2 + 2, y(t) = \sqrt{t}, then your function f(t) = 4 sin(x(t)) cos(y(t)) would be f(t) = 4 sin(t2 + 2) cos(\sqrt{t}), and you could find f'(t) using ordinary differentiation rules.

In your example, x and y are probably very simple functions, possibly constant functions like x(t) = pi/6, y(t) = pi/3 for all t. They may also turn out to be complicated functions like x(t) = sin(t) + pi/2, y(t) = arccos(1/(4sin(sin(t) + pi/2))), but in any case, f(t) = 4 sin(x(t)) cos(y(t)) turns out to be a constant function: f(t) = 1 for all t. Therefore f'(t) = 0.

(Note: differentiating y with respect to x when 4 sinx cosy = 1 would be more complicated, involving implicit differentiation.)

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