What is the derivative of #f(x)=csc^-1(x)# ?
First we will rewrite the equation in a form that is easier to work with.
Take the cosecant of both sides:
Rewrite in terms of sine:
Now, taking the derivative should be easier. It's now just a matter of chain rule.
We know that
So, take the derivative of the outside function, then multiply by the derivative of
The derivative of
Simplifying 8. gives us:
To make the statement a little prettier, we can bring the square of
And there is our answer. Remember, derivatives problems involving inverse trig functions are mostly an exercise in your knowledge of trig identities. Use them to break down the function into a form that's easy to differentiate.