Chapter 12: Q. 72 (page 945)

$Let z = f (x, y) g (x, y) . Show that \frac{\partial z}{\partial x} = \frac{\partial f}{\partial x} g (x, y) + f (x, y) \frac{\partial g}{\partial x} and \frac{\partial z}{\partial y} = \frac{\partial f}{\partial y} g (x, y) + f (x, y) \frac{\partial g}{\partial y}$

Short Answer

Expert verified

$\frac{\partial z}{\partial x} = \frac{\partial f}{\partial x} g (x, y) + f (x, y) \frac{\partial g}{\partial x} \frac{\partial z}{\partial y} = \frac{\partial f}{\partial y} g (x, y) + f (x, y) \frac{\partial g}{\partial y}$

Step by step solution

Step 1. Given information

A function, $z = f (x, y) g (x, y)$

Step 2. Proofs of given partial derivatives

$LHS = \frac{\partial z}{\partial x} = \frac{\partial f (x, y) g (x, y)}{\partial x} Using product rule and definition of partial derivatives, = \frac{\partial f}{\partial x} g (x, y) + f (x, y) \frac{\partial g}{\partial x} Similarly, for partial derivative with respect to y, we get, LHS = \frac{\partial z}{\partial y} = \frac{\partial f (x, y) g (x, y)}{\partial y} Using product rule and definition of partial derivatives, = \frac{\partial f}{\partial y} g (x, y) + f (x, y) \frac{\partial g}{\partial y} = RHS$

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$Let z = f (x, y) g (x, y) . Show that \frac{\partial z}{\partial x} = \frac{\partial f}{\partial x} g (x, y) + f (x, y) \frac{\partial g}{\partial x} and \frac{\partial z}{\partial y} = \frac{\partial f}{\partial y} g (x, y) + f (x, y) \frac{\partial g}{\partial y}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proofs of given partial derivatives

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Letz=f(x,y)g(x,y).Showthat∂z∂x=∂f∂xg(x,y)+f(x,y)∂g∂xand∂z∂y=∂f∂yg(x,y)+f(x,y)∂g∂y

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proofs of given partial derivatives

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

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Study anywhere. Anytime. Across all devices.

$Let z = f (x, y) g (x, y) . Show that \frac{\partial z}{\partial x} = \frac{\partial f}{\partial x} g (x, y) + f (x, y) \frac{\partial g}{\partial x} and \frac{\partial z}{\partial y} = \frac{\partial f}{\partial y} g (x, y) + f (x, y) \frac{\partial g}{\partial y}$