Q 43.

Question

In Exercises $43 - 50$ , compute all of the second-order partial derivatives for the function $f (x, y) = e^{x} \sin (x y)$ and show that the mixed partial derivatives are equal.

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Answer

The second order partial derivatives for the function are

$\frac{\partial^{2} f}{\partial x^{2}} = 2 y e^{x} \cos (x y) - y^{2} e^{x} \sin (x y) + e^{x} \sin (x y) \frac{\partial^{2} f}{\partial y^{2}} = - x^{2} e^{x} \sin (x y)$

1Step 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.

2Step 2. Finding second order partial derivative

$f (x, y) = e^{x} \sin (x y) \dots \dots (1)$

Partially differentiate equation $(1)$ both sides with respect to $x$

$\Rightarrow \frac{\partial f}{\partial x} = y e^{x} c o s (x y) + e^{x} \sin (x y)$

Again partially differentiate both sides with respect to $x$

$\Rightarrow \frac{\partial^{2} f}{\partial x^{2}} = 2 y e^{x} \cos (x y) - y^{2} e^{x} \sin (x y) + e^{x} \sin (x y)$

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

$\Rightarrow \frac{\partial f}{\partial y} = x e^{x} \cos (x y)$

Again partially differentiate both sides with respect to $y$

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

3Step 3. Finding mixed order partial derivative

$f (x, y) = e^{x} \sin (x y)$

$\Rightarrow \frac{\partial^{2} f}{\partial x \partial y} = \frac{\partial}{\partial x} (x e^{x} \cos (x y))$

$\Rightarrow \frac{\partial^{2} f}{\partial x \partial y} = e^{x} \cos (x y) + x e^{x} \cos (x y) - x y e^{x} \sin (x y)$

Also

$\Rightarrow \frac{\partial^{2} f}{\partial y \partial x} = \frac{\partial}{\partial y} (y e^{x} \cos (x y) + e^{x} \sin (x y))$

$\Rightarrow \frac{\partial^{2} f}{\partial y \partial x} = e^{x} \cos (x y) - y x e^{x} \sin (x y) + x e^{x} \cos (x y)$

Now as we observe

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

Q 42.

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