The derivative of tan(x) can be found using the quotient rule since tan(x) = sin(x)/cos(x) as:
(d/dx) tan(x) = [(d/dx) sin(x) cos(x) – sin(x) (d/dx) cos(x)]/cos^2(x)
= [cos^2(x) + sin^2(x)]/cos^2(x) [Applying product rule in first term and differentiating sin and cos gives this equation]
= 1/cos^2(x)
= sec^2(x)
Therefore, (d/dx) tan(x) = sec^2(x).
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