Find gradient of curve y=x(4-x^2)^0.5 where it crosses line y=x

First find the points of interception. That is solve:

x(4-x^2)^0.5 = x which gives

x = 0, sqrt(3), - sqrt(3)

Then differentiate y = x(4-x^2)^1/2 to find dy/dx, the gradient at any point. Using Product Rule we get:

dy/dx = (4-x^2)^(1/2) - x^2/(4-x^2)^(1/2)

Substitute for the three values of x in turn to find the gradient at that point. Thus:

When x = 0, dy/dx = 2

When x = sqrt(3), dy/dx = -2

When x = -sqrt(3), dy/dx = -2