Calculus: why the derivative of tanx is (secx)^2

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Tutoring calculus, everyday formulas students count on can often be verified. The tutor mentions the derivative of tanx.

It’s true that (tanx)’ = (secx)^2. But why?

Realizing tanx = sinx/cosx, then using the the quotient rule (u/v)’ = (vu’ – uv’)/v^2, we get

(sinx/cosx)’ = (cosx*cosx – sinx(-sinx))/(cosx)^2

cosx*cosx + sinx*sinx = 1, giving

(sinx/cosx)’ = 1/(cosx)^2 = (secx)^2

Source:

Larson, Roland E. and Robert P. Hostetler. Calculus. Toronto: D C Heath and Company, 1989.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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