**Problem 45:** Find the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) at the point \((-1, 1, -1)\) for the given implicit function: \[ y e^{y-x^2} + z^2 = -3z - y^3 \] ### Solution Summary: This problem requires the application of implicit differentiation to find the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) for the given function at the specified point. The procedure typically involves differentiating both sides of the equation with respect to \(x\) and \(y\) separately, treating \(z\) as a function of \(x\) and \(y\), and then solving the resulting system of equations. ### Steps to Solve: 1. Differentiate the equation implicitly with respect to \(x\) to find \(\frac{\partial z}{\partial x}\). 2. Differentiate the equation implicitly with respect to \(y\) to find \(\frac{\partial z}{\partial y}\). 3. Substitute the point \((-1, 1, -1)\) into the resulting expressions to find the numerical values of the partial derivatives.