help with higher derivatives please?

Question

how would you find the 1st and 2nd derivative of this function?

h(x)= tan^-1(x^2)

Answer 1

The first derivative of tan^-1(x) is 1/(1+x^2). So, the first derivative of your function (with chain rule) is 2x/(1+x^4), or (2x)(1+x^4)^-1.

The derivative of this derivative is 2/(1+x^4)+(8x^4)/(1+x^4)^2. This combines to (2+10x^4)/(1+x^4)^2.

Answer 2

tan^-1(x^2) = cot(x^2)

Use the chain rule and the derivitive of cot(x) = -csc^2(x)

first derivitive:
-2x*csc^2(x^2)

Use the product rule and chain rule on this to get the 2nd derivitive.