The derivative of tan(x) can be found using the quotient rule of differentiation. The quotient rule states,
d/dx [f(x)/g(x)] = [g(x)f'(x) – f(x)g'(x)] / [g(x)]^2
Applying the quotient rule to the function tan(x), we get:
d/dx[tan(x)] = [sec^2(x) * 1 – tan(x) * sec^2(x) * 1]/[cos^2(x)]
= sec^2(x) – tan(x)sec^2(x)
= sec^2(x) [1 – tan(x)]
Therefore, the derivative of tan(x) is sec^2(x) times the quantity [1 – tan(x)].
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