The derivative of tan(x) can be found using the quotient rule, since tan(x) can be written as sin(x)/cos(x):
f(x) = sin(x)
g(x) = cos(x)
Then, using the quotient rule:
[f'(x)g(x) – g'(x)f(x)] / [g(x)]^2
= [(cos(x))(cos(x)) – (-sin(x))(sin(x))] / [cos(x)]^2
= (cos^2(x) + sin^2(x)) / cos^2(x)
= 1/cos^2(x)
= sec^2(x)
Therefore, the derivative of tan(x) is sec^2(x).
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