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Suppose that the dependent variables z and w are function of the dependent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0 where f_{z}*g_{w} - f_{w}*g_{z} = 1 Which one of the following is correct?
(a) z*{x} = f_{w}*g_{x}* - f_{x}*g_{w}*
(b) z *{x}= f_{x}*g*{w} - f_{w}*g_{x}*
(c) z*{x} = f_{z}*g_{x}* - f_{x}*g_{z}*
(d) z*{x} = f_{z}*g_{w} *- f_{z}*g_{x}*?
Most Upvoted Answer
Suppose that the dependent variables z and w are function of the depen...
Understanding the Problem
In this scenario, we have two equations, f(x, y, z, w) = 0 and g(x, y, z, w) = 0. The derivatives of these functions with respect to z and w are given, and we know that f_z * g_w - f_w * g_z = 1. This condition allows us to use the implicit function theorem to derive relationships between the variables.
Identifying the Correct Option
We want to find the expression for z{x}, representing the partial derivative of z with respect to x. The relationship we can use is derived from the formula for implicit differentiation in multivariable calculus.
Key Relationships
- The formula for finding the derivative of z with respect to x when dealing with implicit functions can be expressed as:
z{x} = (f_x * g_w - f_w * g_x) / (f_z * g_w - f_w * g_z)
- Given that f_z * g_w - f_w * g_z = 1, this simplifies our expression significantly.
Evaluating the Options
- Option (a): z{x} = f_w * g_x - f_x * g_w
- Option (b): z{x} = f_x * g_w - f_w * g_x
- Option (c): z{x} = f_z * g_x - f_x * g_z
- Option (d): z{x} = f_z * g_w - f_z * g_x
Among these, the correct formulation aligns with option (b):
Correct Answer: z{x} = f_x * g_w - f_w * g_x
This choice follows from the implicit differentiation principles and the specific relationship derived from the given condition, confirming it as the valid expression for z{x} with respect to x.
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Suppose that the dependent variables z and w are function of the dependent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0 where f_{z}*g_{w} - f_{w}*g_{z} = 1 Which one of the following is correct?(a) z*{x} = f_{w}*g_{x}* - f_{x}*g_{w}*(b) z *{x}= f_{x}*g*{w} - f_{w}*g_{x}*(c) z*{x} = f_{z}*g_{x}* - f_{x}*g_{z}*(d) z*{x} = f_{z}*g_{w} *- f_{z}*g_{x}*?
Question Description
Suppose that the dependent variables z and w are function of the dependent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0 where f_{z}*g_{w} - f_{w}*g_{z} = 1 Which one of the following is correct?(a) z*{x} = f_{w}*g_{x}* - f_{x}*g_{w}*(b) z *{x}= f_{x}*g*{w} - f_{w}*g_{x}*(c) z*{x} = f_{z}*g_{x}* - f_{x}*g_{z}*(d) z*{x} = f_{z}*g_{w} *- f_{z}*g_{x}*? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Suppose that the dependent variables z and w are function of the dependent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0 where f_{z}*g_{w} - f_{w}*g_{z} = 1 Which one of the following is correct?(a) z*{x} = f_{w}*g_{x}* - f_{x}*g_{w}*(b) z *{x}= f_{x}*g*{w} - f_{w}*g_{x}*(c) z*{x} = f_{z}*g_{x}* - f_{x}*g_{z}*(d) z*{x} = f_{z}*g_{w} *- f_{z}*g_{x}*? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose that the dependent variables z and w are function of the dependent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0 where f_{z}*g_{w} - f_{w}*g_{z} = 1 Which one of the following is correct?(a) z*{x} = f_{w}*g_{x}* - f_{x}*g_{w}*(b) z *{x}= f_{x}*g*{w} - f_{w}*g_{x}*(c) z*{x} = f_{z}*g_{x}* - f_{x}*g_{z}*(d) z*{x} = f_{z}*g_{w} *- f_{z}*g_{x}*?.
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