Verified answer

Sometimes, a change of variable helps. Let r be a real variable not related to anything.

Let f(r) = sqrt(r) and g(r) = 36 - 9*r^2 and h(r) = sin r. The derivatives with respect to r are:

f '(r) = 1/(2*sqrt(r)), g'(r) = -18r, h'(r) = cos r.

Then S = f(g(h(x))), right? The chain rule tells you that S' is:

S'(x) = f '(g(h(x))) * [g(h(x))]'

That left factor is (df/dr) evaluated at r=g(h(x)), NOT (df/dx). The right factor is (d/dx)g(h(x)), which you can get with a second chain rule as [g(h(x))]' = g'(h(x)) * h'(x). The first factor of that is (dg/dr) evaluated at r=h(x), NOT dg/dx.

S'(x) = f '(g(h(x))) * g'(h(x)) * h'(x)

= 1/(2*sqrt(36 - 9*sin^2 x)) * -18*sin x * cos x

Simplify to taste.

If the closed bracket is after the x

then -(9*cos(x)*sin(x))/sqrt(36-9*sin(x)^2)

is a solution because integrating it

returns sqrt(36-9*sin(x)^2).