Chapter 12: Q. 76 (page 946)

$Let f (x) be a differentiable function of x, g (y) be a differentiable function of y, and h (x, y) = f (x) g (y) . Prove that \frac{\partial^{2} h}{\partial x \partial y} = \frac{\partial^{2} h}{\partial y \partial x} .$

Short Answer

Expert verified

$\frac{\partial^{2} h}{\partial x \partial y} = \frac{\partial^{2} h}{\partial y \partial x}$

Step by step solution

Step 1. Given information

$h (x, y) = f (x) g (y) Where, f (x) i s a differentiable function of x, g (y) i s a differentiable function of y .$

Step 2. Proof of given partial derivative

$LHS = \frac{\partial^{2} h}{\partial x \partial y} = \frac{\partial}{\partial x} (\frac{\partial f (x) g (y)}{\partial y}) = \frac{\partial f (x) g' (y)}{\partial x} = f' (x) g' (y) RHS = \frac{\partial^{2} h}{\partial y \partial x} = \frac{\partial}{\partial y} (\frac{\partial f (x) g (y)}{\partial x}) = \frac{\partial f' (x) g (y)}{\partial y} = f' (x) g' (y) LHS = RHS$

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$Let f (x) be a differentiable function of x, g (y) be a differentiable function of y, and h (x, y) = f (x) g (y) . Prove that \frac{\partial^{2} h}{\partial x \partial y} = \frac{\partial^{2} h}{\partial y \partial x} .$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proof of given partial derivative

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Letf(x)beadifferentiablefunctionofx,g(y)beadifferentiablefunctionofy,andh(x,y)=f(x)g(y).Provethat∂2h∂x∂y=∂2h∂y∂x.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proof of given partial derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

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Study anywhere. Anytime. Across all devices.

$Let f (x) be a differentiable function of x, g (y) be a differentiable function of y, and h (x, y) = f (x) g (y) . Prove that \frac{\partial^{2} h}{\partial x \partial y} = \frac{\partial^{2} h}{\partial y \partial x} .$