Chapter 13.5, Problem 49ES

(a) Suppose that $z=f\left(u\right)\text{and}u=g\left(x,y\right).$ Draw a tree diagram, and use it to construct chain rules that express $\partial z/\partial x\text{and}\partial z/\partial y$ in terms of $dz/du,\partial u/\partial x\text{and}\partial u/\partial y.$

(b) show that

$\begin{array}{l}\frac{{\partial }^{2}z}{\partial x^2}=\frac{dz}{du}\frac{{\partial }^{2}u}{\partial x^2}+\frac{d^{2}z}{d{u}^2}{\left(\frac{\partial u}{\partial x}\right)}^2\\ \frac{{\partial }^{2}z}{\partial y^2}=\frac{dz}{du}\frac{{\partial }^{2}u}{\partial y^2}+\frac{d^{2}z}{d{u}^2}{\left(\frac{\partial u}{\partial y}\right)}^2\\ \frac{{\partial }^{2}z}{\partial y\partial x}=\frac{dz}{du}\frac{{\partial }^{2}u}{\partial y\partial x}+\frac{d^{2}z}{d{u}^2}\frac{\partial u}{\partial x}\frac{\partial u}{\partial y}\end{array}$