Yeah for Google+!!! After posting yesterday's missive about figuring out the origin of the secant squared term, mathematician extraordinare and Google+er +John Baez pointed out to me that the derivative of tangent is secant squared and that I need not have fussed so much with the geometry. John said:

I wouldn't call the introduction of sec^2(θ) a "substitution". Introducing θ in the first place counts as a substitution, since you're trying to simplify an integral by replacing some other variable with this one. But it's just a mathematical fact that the derivative of tanθ is sec^2(θ), so

d tan(θ) = sec^2(θ) dθ

I would use this equation automatically and unthinkingly, but it looks like you're trying to find a geometrical explanation of this equation. If you're figuring it out for yourself because nobody told you the derivative of tan(θ) is sec^2(θ), that's very laudable! The standard approach is to express tan in terms of sin and cos, and use the quotient rule for derivatives.

I guess the lesson is when you see differential on both sides of an expression, think calculus, not geometry!

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