Chapter 12: Q. 73 (page 945)

$Let z = f (x, y) g (x, y) . Show that \frac{\partial z}{\partial x} = \frac{f_{x} (x, y) g (x, y) - f (x, y) g_{x} (x, y)}{(g {(x, y))}^{2}} and \frac{\partial z}{\partial y} = \frac{f_{y} (x, y) g (x, y) - f (x, y) g_{y} (x, y)}{(g {(x, y))}^{2}}$

Short Answer

Expert verified

$\frac{\partial z}{\partial x} = \frac{f_{x} (x, y) g (x, y) - f (x, y) g_{x} (x, y)}{(g {(x, y))}^{2}} \frac{\partial z}{\partial y} = \frac{f_{y} (x, y) g (x, y) - f (x, y) g_{y} (x, y)}{(g {(x, y))}^{2}}$

Step by step solution

Step 1. Given information

A function, $z = \frac{f (x, y)}{g (x, y)}$

Step 2. Proofs of given partial derivatives

$L H S = \frac{\partial z}{\partial x} = \frac{\partial \frac{f (x, y)}{g (x, y)}}{\partial x} Using quotient rule and definition of partial derivatives, = \frac{f_{x} (x, y) g (x, y) - f (x, y) g_{x} (x, y)}{(g {(x, y))}^{2}} Similarly, for the partial derivative with respect to y, we get, LHS = \frac{\partial z}{\partial y} = \frac{\partial \frac{f (x, y)}{g (x, y)}}{\partial y} Using quotient rule and definition of partial derivatives, = \frac{f_{y} (x, y) g (x, y) - f (x, y) g_{y} (x, y)}{(g {(x, y))}^{2}}$

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$Let z = f (x, y) g (x, y) . Show that \frac{\partial z}{\partial x} = \frac{f_{x} (x, y) g (x, y) - f (x, y) g_{x} (x, y)}{(g {(x, y))}^{2}} and \frac{\partial z}{\partial y} = \frac{f_{y} (x, y) g (x, y) - f (x, y) g_{y} (x, y)}{(g {(x, y))}^{2}}$

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=fx(x,y)g(x,y)−f(x,y)gx(x,y)(g(x,y))2and∂z∂y=fy(x,y)g(x,y)−f(x,y)gy(x,y)(g(x,y))2

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

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$Let z = f (x, y) g (x, y) . Show that \frac{\partial z}{\partial x} = \frac{f_{x} (x, y) g (x, y) - f (x, y) g_{x} (x, y)}{(g {(x, y))}^{2}} and \frac{\partial z}{\partial y} = \frac{f_{y} (x, y) g (x, y) - f (x, y) g_{y} (x, y)}{(g {(x, y))}^{2}}$