Derivative Question

Question

Let f(x) = (arcsin)(sqrt((1-sinx)/(1+sinx))), find f’(0)

Answer 1

My recommendation would be to rewrite the function implicitly, as follows.

After recasting the equation as "y = ... ", take the sine of both sides.

We'll now have something of the form "sin y = sqrt( ... ) ".

Continuing with the simplification by squaring both sides we get sin² y = (1 - sin x)/(1 + sin x).

This may help when taking the derivative, assuming you are comfortable with the implicit differentiation.

Also, before continuing with that step, it may be helpful to rewrite the right side of the above as follows:

(1- sin x)/(1 + sin x) = (1 + sin x - sin x - sin x) / (1 + sin x) = (1 + sin x)/(1 + sin x) - 2 sin x/(1 + sin x) = ...

... = 1 - 2 sin x/(1 + sin x) = 1 - 2/(csc x + 1).

We'd now have the following equivalent implicit form of the function from the start:

sin² y = 1 - 2 / ( csc x + 1 )

From here, implicit differentiation will get the job done. Let me know if I can help with that, or if you'd like a more direct approach, where we will not attempt to avoid the chain rule and derivatives of inverse trig functions and roots.

Answer 2

Undefined

Let u = sqrt[(1 - sin(x))/(1 + sin(x))]

Then f(u) = arcsin(u) so f '(u) = [1/sqrt(1 - u²)]•u' but sqrt(1 - u²) at x = 0 is equal to zero so we get a "divide by zero" condition.