Course:MATH200/HigherOrderChainRule

Chain rule for second order partial derivatives

Using the chain rule to compute second-order partial derivatives can be a little bit confusing. It helps to see an example to clarify it. This example is part of a problem in our text.

Example

Let z=f(x,y) be a function with continuous second-order partial derivatives. Suppose that

$\displaystyle x=e^{s}\cos t,y=e^{s}\sin t$

Compute ${\frac {\partial ^{2}z}{\partial s^{2}}}$.

Solution

By the chain rule we have

${\frac {\partial z}{\partial s}}={\frac {\partial z}{\partial x}}{\frac {\partial x}{\partial s}}+{\frac {\partial z}{\partial y}}{\frac {\partial y}{\partial s}}=e^{s}\cos t{\frac {\partial z}{\partial x}}+e^{s}\sin t{\frac {\partial z}{\partial y}}.$

Now we have

 ${\frac {\partial ^{2}z}{\partial s^{2}}}={\frac {\partial }{\partial s}}{\frac {\partial z}{\partial s}}=e^{s}\cos t{\frac {\partial z}{\partial x}}+e^{s}\cos t{\frac {\partial }{\partial s}}{\frac {\partial z}{\partial x}}+e^{s}\sin t{\frac {\partial z}{\partial y}}+e^{s}\sin t{\frac {\partial }{\partial s}}{\frac {\partial z}{\partial y}}.$

We have to expand out the second and fourth terms now. We start by expanding

${\frac {\partial }{\partial s}}{\frac {\partial z}{\partial x}}={\frac {\partial ^{2}z}{\partial x^{2}}}{\frac {\partial x}{\partial s}}+{\frac {\partial ^{2}z}{\partial y\partial x}}{\frac {\partial y}{\partial s}}=e^{s}\cos t{\frac {\partial ^{2}z}{\partial x^{2}}}+e^{s}\sin t{\frac {\partial ^{2}z}{\partial y\partial x}}.$

Similarly

${\frac {\partial }{\partial s}}{\frac {\partial z}{\partial y}}={\frac {\partial ^{2}z}{\partial x\partial y}}{\frac {\partial x}{\partial s}}+{\frac {\partial ^{2}z}{\partial y^{2}}}{\frac {\partial y}{\partial s}}=e^{s}\cos t{\frac {\partial ^{2}z}{\partial x\partial y}}+e^{s}\sin t{\frac {\partial ^{2}z}{\partial y^{2}}}.$

Now we substitute to get

${\frac {\partial ^{2}z}{\partial s^{2}}}=e^{s}\cos t{\frac {\partial z}{\partial x}}+e^{2s}\cos ^{2}t{\frac {\partial ^{2}z}{\partial x^{2}}}+e^{2s}\cos t\sin t{\frac {\partial ^{2}z}{\partial y\partial x}}+e^{s}\sin t{\frac {\partial z}{\partial y}}+e^{2s}\sin t\cos t{\frac {\partial ^{2}z}{\partial x\partial y}}+e^{2s}\sin ^{2}t{\frac {\partial ^{2}z}{\partial y^{2}}}.$

Finally, we can simplify this a little bit using Clairaut's formula to

${\frac {\partial ^{2}z}{\partial s^{2}}}=e^{s}\cos t{\frac {\partial z}{\partial x}}+e^{s}\sin t{\frac {\partial z}{\partial y}}+e^{2s}\cos ^{2}t{\frac {\partial ^{2}z}{\partial x^{2}}}+e^{2s}\sin ^{2}t{\frac {\partial ^{2}z}{\partial y^{2}}}+2e^{2s}\sin t\cos t{\frac {\partial ^{2}z}{\partial x\partial y}}.$