Question

Use Implicit Differentiation to calculate the partial derivative ∂z/∂x in equation x^2y + y^2z + xz^2 = 10

Answer 1

$$\frac\partial{\partial x}\left[x^2y+y^2z+xz^2\right]=\frac\partial{\partial x}[10]\\2xy+x^2\frac{\partial y}{\partial x}+2yz\frac{\partial y}{\partial x}+y^2\frac{\partial z}{\partial x}+z^2+2xz\frac{\partial z}{\partial x}=0\\\frac{\partial z}{\partial x}\left(y^2+2xz\right)=-\left(2xy+x^2\frac{\partial y}{\partial x}+2yz\frac{\partial y}{\partial x}+z^2\right)\\\frac{\partial z}{\partial x}=-\frac{2xy+x^2\frac{\partial y}{\partial x}+2yz\frac{\partial y}{\partial x}+z^2}{y^2+2xz}$$Now, I believe they want \(y\) to not be a function of \(x\), so \(\dfrac{\partial y}{\partial x}=0\). Now we're left with:$$\frac{\partial z}{\partial x}=-\frac{2xy+z^2}{y^2+2xz}$$