Show that the derivative of tan(x) is sec^2(x), where sec(x) is defined as 1/cos(x). [Hint: think of tan(x) as a quotient of two related functions and apply the appropriate identity]
tan(x) is defined as sin(x)/cos(x) For a function which can be written as f(x) = u(x)/v(x) the quotient rule can be appliedThe quotient rule states f ' (x) = (u'v-v'u)/v^2 Applying this to the formula for tan we obtain ( cos(x)cos(x) - (-sin(x)sin(x))/(cos(x)^2)Examining the numerator the minus's cancel and we obtain sin(x)^2+cos(x)^2 by a quotable identity this always equals 1Hence the expression simplifies to 1/cos(x)^2 = sec^2(x) as originally statedHence shown.
OC
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