### An annoying derivative

Posted:

**Sun Feb 26, 2017 10:07 pm UTC**I’m trying to take the derivative of a messy function at x=1. I’m pretty sure that y′(1) = 1, but I’m having a world of difficulty proving it. Here’s the setup:

t(x) = (2/π) · arcsin( (2/π) · ( arcsin(x) + x · √(1 - x²) ) )

y(x) = t · √(1 - t²) + x · (1 - √(1 - t²))

Now, I’m reasonably confident that the following are correct:

dt/dx = (8 · √(1 - x²)) / (π² · cos((π/2) · t))

dy/dx = 1 - √(1 - t²) + (dt/dx) · (1 + q·t - 2t²) / √(1 - t²)

It’s the next bit, trying to evaluate dy/dx at x=1, where I keep getting tangled up. When I applied L’Hôpital’s rule it just seemed to open a rabbit hole.

t(x) = (2/π) · arcsin( (2/π) · ( arcsin(x) + x · √(1 - x²) ) )

y(x) = t · √(1 - t²) + x · (1 - √(1 - t²))

Now, I’m reasonably confident that the following are correct:

dt/dx = (8 · √(1 - x²)) / (π² · cos((π/2) · t))

dy/dx = 1 - √(1 - t²) + (dt/dx) · (1 + q·t - 2t²) / √(1 - t²)

It’s the next bit, trying to evaluate dy/dx at x=1, where I keep getting tangled up. When I applied L’Hôpital’s rule it just seemed to open a rabbit hole.