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danerape
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How does one go about rigorously proving that (dy/dx)=1/(dx/dy)?
Thanks
Thanks
danerape said:...what if I can't solve y=f(x) for x as a function of y explicitly?
(dy/dx) is a mathematical notation used to represent the derivative of a function y with respect to the independent variable x. It is a measure of the rate of change of y with respect to x.
(dx/dy) is a mathematical notation used to represent the derivative of a function x with respect to the dependent variable y. It is a measure of the rate of change of x with respect to y.
This is a result of the inverse relationship between derivatives. The derivative of a function y with respect to x is the reciprocal of the derivative of x with respect to y.
To prove this, we can use the chain rule and implicit differentiation to show that the derivative of y with respect to x is equal to the derivative of x with respect to y multiplied by the derivative of y with respect to x. Simplifying this expression leads to (dy/dx) = 1/(dx/dy).
This relationship is useful in various fields of science and engineering, such as physics, economics, and engineering. It can be used to analyze rates of change, optimize functions, and solve differential equations.