Question

How do you differentiate # g(x) = sqrt(arc csc(x^3+1) #?

Answer 1

Supposing `|(x^3+1)|>1`, we can proceed using the specific rule for `arccsc(f(x))` and also chain rule for the square root.

Explanation:

• `(arccscu)'=(-u')/(|u|sqrt(u^2-1))`

• Chain rule: `(dy)/(dx)=(dy)/(dv)(dv)/(dx)`

Renaming `v=arccsc(x^3+1)`, we have `g(x)=sqrt(v)`

`(dg(x))/(dx)=1/(2v)((-3x^2)/(|x^3+1|sqrt(x^3(x^3+2))))=((-3x^2)/(arccsc(x^3+1)|x^3+1|sqrt(x^3(x^3+2))))`