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

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

Q. 73 - Calculus | StudyQuestionHub

Q. 73

Question

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

Step-by-Step Solution

Verified

Answer

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

1Step 1. Given information

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

2Step 2. Proofs of given partial derivatives

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

Q. 72

TF. 1

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