1. Let's start by stating the problem: Find the partial derivative of a given function with respect to a specified variable. 2. Suppose the function is $f(x,y) = x^2 y + 3xy^2$ and we want to find $\frac{\partial f}{\partial x}$. 3. To find $\frac{\partial f}{\partial x}$, treat $y$ as a constant and differentiate with respect to $x$: $$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (x^2 y + 3xy^2)$$ 4. Differentiating term-by-term: - For $x^2 y$, derivative is $2x y$ because $y$ is constant. - For $3xy^2$, derivative is $3 y^2$ because $y^2$ is constant. 5. Therefore, $$\frac{\partial f}{\partial x} = 2 x y + 3 y^2$$ 6. This is the partial derivative of the function with respect to $x$. The final answer is: $$\frac{\partial f}{\partial x} = 2 x y + 3 y^2$$