Question

I've done everything I can think of to solve this, but I have no idea. Find the derivative: y = sq rt{11x + sqrt[11x + sqrt(11x)]}

Answer 1

I think you mean

y = √(11x + √(11x+√(11x)))
y^2 = 11x + √(11x+√(11x))
If u^2 = 11x,
u' = 11/(2√(11x))
y^2 = u^2 + √(u^2+u)
2y y' = 2u u' + (2u+1)u'/(2√(u^2+u))
2y y' = u' (4u+1)/(2√(u^2+u))

Now substitute back in for y, u, u' and you will get the mess shown here:

http://www.wolframalpha.com/input/?i=derivative+%E2%88%9A%2811x+%2B+%E2%88%9A%2811x%2B%E2%88%9A%2811x%29%29%29

Problems like this are kinda fun, but they really don't help you learn any calculus -- just keeping track of algebraic details.