Q 47. - Calculus

Question

In Exercises $43-50$, compute all of the second-order partial derivatives for the functions $f\left(x,y\right)=x\sin y$ and show that the mixed partial derivatives are equal. 

Step-by-Step Solution

Answer

The second order partial derivatives for the function are

$$\begin{gathered} \frac{{\partial }^{2}f}{\partial x^2}=0 \\ \frac{{\partial }^{2}f}{\partial y^2}=-x\sin y \end{gathered}$$

Step 1. Definition

Clairaut's theorem on equality of mixed partials states that under assumption of continuity of both the second-order mixed partials of a function of two variables, the two mixed partials are equal. 

Step 2. Finding second order partial derivative

$f\left(x,y\right)=x\sin y \cdots \cdots \left(1\right)$
Partially differentiate equation $\left(1\right)$ both sides with respect to $x$

$\Rightarrow \frac{\partial f}{\partial x}=\sin y$

Again partially differentiate both sides with respect to $x$

$\Rightarrow \frac{{\partial }^{2}f}{\partial x^2}=0$

Partially differentiate equation $\left(1\right)$ both sides with respect to $y$

$\Rightarrow \frac{\partial f}{\partial y}=x\cos y$

Again partially differentiate both sides with respect to $y$

$\Rightarrow \frac{{\partial }^{2}f}{\partial y^2}=-x\sin y$

Step 3. Finding mixed order partial derivative

$f\left(x,y\right)=x\sin y$

$$\begin{gathered} \Rightarrow \frac{{\partial }^{2}f}{\partial x\partial y}=\frac{\partial }{\partial x}\left(x\cos y\right) \\ \Rightarrow \frac{{\partial }^{2}f}{\partial x\partial y}=\cos y \end{gathered}$$

Also

$$\begin{gathered} \Rightarrow \frac{{\partial }^{2}f}{\partial y\partial x}=\frac{\partial }{\partial y}\left(\sin y\right) \\ \Rightarrow \frac{{\partial }^{2}f}{\partial y\partial x}=\cos y \end{gathered}$$

Now as we observe

$\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{{\partial }^{2}f}{\partial y\partial x}$ [Hence proved]