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One technique for finding the derivative of the sin function is to find the derivative of arcsin first. The arcsin function can be expressed as the area of a ce

One technique for finding the derivative of the sin function is to find the derivative of arcsin first. The arcsin function can be expressed as the area of a certain figure, and therefore admits an expression as an integral. From this, the derivative of arcsin is immediate. Finally, apply your technique to arcsin(sin(x)) = x and obtain the derivative of sin. A similar technique finds the derivative of exp(x) from ln(x), by defining the latter as the integral of 1/x.

selimthegrim 4y ago
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- ogogmad 4y ago
  :P See https://math.stackexchange.com/questions/75130/how-to-prove-that-lim-limits-x-to0-frac-sin-xx-1/1230225#1230225 https://math.stackexchange.com/questions/75130/how-to-prove-... The OP uses the derivative of sin to find the derivative of arcsin. My point was that the other way round is sort of natural as well. Thought I'd share. The broader point is that you can obtain all the classic definitions and identities of the elementary functions [1] by combining only (i) the four arithmetic operations, (ii) integration, and (iii) taking inverses of functions (iv) the constants 0 and 1. I guess that's a bit cool. This works fine over the real numbers. Over the complex numbers, this might require Riemann surfaces to work. e^z in complex analysis is usually defined by its Taylor series, instead of as the inverse function of the natural log (whose domain is actually a Riemann surface). [1] - https://en.wikipedia.org/wiki/Elementary_function https://en.wikipedia.org/wiki/Elementary_function