4g. Find the first and second derivative.

This is a composed function, so we can use the chain rule to find
its derivative. Fortunately, we know from Problem 4e that the
derivative of tan x is 1/cos^{2}x. After we apply this to the
inside function which is the inverse sine function, we need to
multiply this by the derivative of the inside function. For this we
need to know that

If we put all of this together, we get

This can be simplified

It is not too surprising that this function has an algebraic derivative, because the original function could have been simplified

We can see this either algebraically, as above, or geometrically.

In the picture, y = sin a so a = sin^{-1}y.
tan(sin^{-1}y) =

We can differentiate this fairly easily using the quotient rule and problem 4c.

Finding common denominators to add the fractions on top gives us

There are often many ways to do a calculus problem, and many of those ways will result in answers that look radically different. However, there is a lot of fun to be had showing that while the different answers do not look like they will give you the same thing, they really do.

For the second deritave, it would be easiest to differentiate this last result

For this we use the chain rule

This will clean up to

One would suspect that there is undoubtedly a great deal of fun to be had showing that we get the same result using the other form of the first derivative.

Let us rewrite this as

so that we can use the product rule.

This will clean up to

The same answer we got the other way.