I liked this simple calculus exercise • Hacker News Board

Re: I liked this simple calculus exercise

#161

Post by Spivak » Tue, Apr 18, 2023, 10:33 PM UTC

justeleblanc wrote: post #123

ryan-duve wrote: post #2

This reminds me of an exercise I'll never forget from my Math Methods course: finding the derivative of arcsin(x). It seems almost impossible because, just looking at it, there seems to be nothing you can do to simplify it. Then, out of sheer nothing-else-to-do-ism, you take the sin() of it and realize sin(arcsin(x)) = x. Take the derivative of both sides, apply chain rule and draw a right triangle and you have the a…

That's how you find any reciprocal function's derivative. If g = f^{-1}, then g'(y) = 1/f'(g(y)).

By your rule

    g = arcsin
    f = sin

    g'(x) = 1 / cos(arcsin(x))

But that's not the usual form. Can you show that the denominator is sqrt(1 - x^2)? Doing that requires you make the same observation as in the parent post.

Re: I liked this simple calculus exercise

#162

Post by Tainnor » Wed, Apr 19, 2023, 9:52 AM UTC

Spivak wrote: post #161

justeleblanc wrote: post #123

Earlier quoted context omitted.

That's how you find any reciprocal function's derivative. If g = f^{-1}, then g'(y) = 1/f'(g(y)).

By your rule g = arcsin f = sin g'(x) = 1 / cos(arcsin(x)) But that's not the usual form. Can you show that the denominator is sqrt(1 - x^2)? Doing that requires you make the same observation as in the parent post.

> Can you show that the denominator is sqrt(1 - x^2)?

That's a trig identity that is easily derivable: draw a right triangle with hypotenuse 1 and one side being x. Its one angle will be arcsin(x) by definition. Then by Pythagoras the remaining side is sqrt(1-x^2), which is the cosine of that angle.

I guess GP's point was that the implicit differentiation trick by using the inverse is generalisable and not only restricted to trigonometric functions. Obviously, which simplifications you can then additionally make will depend on the function in question.

Re: I liked this simple calculus exercise

#163

Post by hgsgm » Thu, Apr 20, 2023, 1:49 AM UTC

mjd wrote: post #86

hgsgm wrote: post #77

Earlier quoted context omitted.

You don't need graphs or area at all, of course. All you need is to notice that the frac(x/2) expression is a periodic function that is only integrable piecewise, and is easy to integrate with a change of variables for each piece or a handwave of such. It's actually gnarly to write out a formal proof as a new student would do (it requires principle of induction to handle all the pieces), but easy for an expert to bre…

> All you need is to notice… All you need is a magical inspiration from out of nowhere! But if you don't happen to have that magical inspiration, the graph will make the periodicity visually obvious.

Deciding to draw a graph is also magical inspiration.