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/cos2x. 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-1y. tan(sin-1y) =
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.