Final answer:
The value of y for which the derivative of y with respect to x does not exist is y = 1/e.
Step-by-step explanation:
The given curve is y2^y = x. We need to find the value of y, if any, for which the derivative of y with respect to x does not exist.
To find the derivative of y with respect to x, we need to differentiate both sides of the equation with respect to x.
By taking the natural logarithm of both sides, we can simplify the equation: yln(y) = ln(x).
Now, we can use implicit differentiation to find the derivative: dy/dx * ln(y) + y * (1/y) * dy/dx = 1/x.
Simplifying further, we have: dy/dx * ln(y) + dy/dx = 1/x.
Finally, we can factor out dy/dx: dy/dx * (ln(y) + 1) = 1/x.
The derivative of y with respect to x does not exist when the factor (ln(y) + 1) equals zero. Therefore, when ln(y) + 1 = 0, the derivative does not exist. Solving for y, we have ln(y) = -1, which gives us y = 1/e.