Find the gradients of the curve $\displaystyle y=x\sqrt{4-x^2}$ at the points where it crosses the straight line $\displaystyle y=x$
I am becoming concerned about people posting problems without showing even an attempt at the problem. Write $\displaystyle y= x\sqrt{4- x^2}= x(4- x^2)^{1/2}$ and use the product rule, that you refer to and the chain rule on $\displaystyle (4- x^2)^{1/2}$. Since the problem asks for the derivate where the curve crosses y= x, you will need to solve $\displaystyle y= x= x\sqrt{4- x^2}$ to determine the x values.