Question

nth derivative of tan inverse 2x/1_x^2​

Answer 1

Answer:

As a preliminary:

tan(θ+θ)=2tanθ1−tan2θ

Setting tanθ=x we get:

tan2θ=2x1−x2

2θ=tan−1(2x1−x2)

tan−1(2x1−x2)=2tan−1x

So we are being asked for the n-th derivative of 2tan−1x .

We are asked to find the answer in terms of r and θ but we are not given any definition of these terms. The natural definitions would seem to be:

tanθ=x and

r2=1+x2

We are not told what to differentiate with respect to. Finding the n-th derivative with respect to x is too hard for me. The first derivative is 21+x2 but after that I think it just gets too messy.

So I'll take the easy way out and find the n-th derivative with respect to θ . This is super easy because the required expression is just 2θ . So the first derivative is 2 and all the rest are zero. Job done!

I hope that helps. Let me know if you interpret the question in a different way.