inverse function

Question

Find (f⁻¹)'(a).

f(x) = 3x³ + 2 sin x + 3 cos x,    a = 3

I got zero but it is wrong.

Answer 1

When you take the derivative of the inverse of each piece, each ends up being undefined. So your whole answer would be undefined.

(f^-1)'(x)=1/(9(x/3)^1/3)²+1/(2cos(arcsin(x/2)))-1/(3sin(arccos(x/3)))

When you substitute 3 in the second and third pieces are undefined, since 3/2 is outside the domain of arcsin and sin(arccos(0))=0.

So you would get undefined not zero.