Find the first derivative of f(x) = tan(x).
First, use the trigonometric identity tan(x) = sin(x) / cos(x) to rewrite the function. It should now be very apparent that f(x) is now actually a ratio of two functions, and therefore the quotient rule will be required to differentiate it. Split f(x) into the two functions, g(x) = sin(x) as the numerator, and h(x) = cos(x) as the denominator. Now differentiate g(x) and h(x) with respect to x, giving you g'(x) = cos(x) and h'(x) = -sin(x). Then, using the chain rule, f'(x) = (g'(x)*h(x) - g(x)*h'(x)) / (h(x)2), substitute in the values. This will give you: f'(x) = (cos2(x) + sin2(x)) / cos2(x). While this is the correct answer, it needs to be simplified further using the trig identity cos2(x) + sin2(x) = 1.Therefore the final answer is: f'(x) = 1 / cos2(x) = sec2(x).
