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)]}
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.
To find the derivative of the function y = √{11x + √[11x + √(11x)]}, we can use the chain rule and power rule.
First, let's simplify the expression. Notice that we have nested square roots, so we can rewrite the function as follows:
y = (11x + √(11x + √(11x)))^(1/2)
Now, let's denote the innermost part of the expression as u:
u = 11x
Next, let's find the derivative of u with respect to x:
du/dx = 11
Now, let's find the derivative of the nested square root expression:
v = √(u + √u)
To differentiate v, we need to apply the chain rule. Let's denote the innermost part of v as w:
w = u + √u
Using the power rule and chain rule, we can find the derivative of w with respect to u:
dw/du = 1 + 1/(2√u)
Finally, let's differentiate v with respect to x:
dv/dx = dv/du * du/dx
dv/dx = (dw/du) * (du/dx)
Now, we have all the pieces needed to find the derivative of y:
dy/dx = (1/2) * (11x + √(11x + √(11x)))^(-1/2) * (1 + 1/(2√(11x)))
Therefore, the derivative of y with respect to x is:
dy/dx = (11x + √(11x + √(11x)))^(-1/2) * (1 + 1/(2√(11x))) / 2