Answer
$$\frac{d}{dx}(x^2+y^2)^{3/2}= 3(x+y\frac{dy}{dx})(x^2+y^2)^{1/2} .$$
Work Step by Step
Recall that $(x^n)'=nx^{n-1}$
Using the chain rule, we have
$$\frac{d}{dx}(x^2+y^2)^{3/2}= \frac{3}{2}(x^2+y^2)^{1/2}(2x+2y\frac{dy}{dx})=3(x+y\frac{dy}{dx})(x^2+y^2)^{1/2} .$$
