(partial y)/(partial y)(x^2+y^2)^2

To find the partial derivative of \( y \) with respect to \( y \) in the expression \( (x^2 + y^2)^2 \), we can use the chain rule. Let's break it down step by step.<br /><br />Given:<br />\[ f(y) = (x^2 + y^2)^2 \]<br /><br />We need to find:<br />\[ \frac{\partial f}{\partial y} \]<br /><br />First, let's rewrite the function in terms of \( y \):<br />\[ f(y) = (x^2 + y^2)^2 \]<br /><br />Now, apply the chain rule. The outer function is \( u^2 \) and the inner function is \( x^2 + y^2 \).<br /><br />1. Differentiate the outer function with respect to the inner function:<br />\[ \frac{d}{du}(u^2) = 2u \]<br /><br />2. Differentiate the inner function with respect to \( y \):<br />\[ \frac{d}{dy}(x^2 + y^2) = 2y \]<br /><br />Now, combine these results using the chain rule:<br />\[ \frac{\partial f}{\partial y} = 2(x^2 + y^2) \cdot 2y \]<br /><br />Simplify:<br />\[ \frac{\partial f}{\partial y} = 4y(x^2 + y^2) \]<br /><br />So, the partial derivative of \( (x^2 + y^2)^2 \) with respect to \( y \) is:<br />\[ \boxed{4y(x^2 + y^2)} \]