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NanoMath
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In the proof of uniqueness of ( multi-variable ) derivative from Rudin, I am a little stuck on why the inequality holds. Rest of the proof after that is clear .
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"Proof of derivative uniqueness" is a mathematical concept that states that if a function has a derivative at a certain point, then that derivative is unique and cannot have any other value. This concept is important in calculus and is often used to prove mathematical theorems.
"Proof of derivative uniqueness" is important because it allows us to make precise mathematical statements and solve problems involving rates of change. It also helps us understand the behavior of functions and their derivatives, which is essential in many areas of science and engineering.
"Proof of derivative uniqueness" is proven using the Mean Value Theorem, which states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists a point within the interval where the derivative of the function equals the slope of the secant line connecting the endpoints of the interval.
Yes, "Proof of derivative uniqueness" can be applied to all continuous and differentiable functions. However, it may not be applicable to functions that are not continuous or differentiable at certain points.
"Proof of derivative uniqueness" has many real-world applications, such as in physics and engineering where it is used to calculate rates of change and solve optimization problems. It is also used in economics to analyze supply and demand curves, and in biology to model population growth and decay.