Differentiation Variable

In summary, the conversation discusses finding the directional derivative of a function with respect to a change in the ratio of two variables. The solution involves using the vector orthogonal to the ratio and setting one of the variables as a function of the other. The final result is the derivative of the function with respect to the chosen variable.
  • #1
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I'm working out some problems, and I'm ending up with a term similar to the following:

du/d(y/u)

I'm differentiating with respect to y/u. Both y and u are variables. How can I divide that up to represent differentiation with just one variable (Even if it means expanding the term)?

Is it mathematically possible to do that?

Thanks.
 
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  • #2
I'm really sure what you're trying to do, but if you have a function [tex]f(x,y)[/tex] and you want to know how [tex]f[/tex] changes in the limiting case with respect to a change in the ratio of [tex]y/x[/tex], then this is a case for the directional derivative. Note that [tex]v = (x, y)[/tex] is the vector where [tex]y/x[/tex] stays constant, so you'd want to take the vector orthogonal to that; ie, use [tex]v = (y, -x)[/tex]. In that case, [tex]\nabla_v f(x,y) = \nabla f(x,y) \cdot v[/tex]
 
  • #3
Let v= u/y. Then, if f(u,y) is any function of u and y, [itex]df/dv= \partial f/\partial u \partial u\partial v+ \partial f/\partial y \partial y/partial v[itex].

Since, here, v= u/y, so u= yv and [itex]\partial u/\partial v= y[itex]. Similarly, y= u/v so [itex]\partial y/\partial v= -u/v^2= -u/(u^2/y^2)= -y^2/u[/itex].

That is, [itex]df/dv= y\partial f/\partial u- (y^2/u)\partial f/\partial y[/itex]

And, since here f(u,y)= u, [itex]\partial f/\partial u= 1[/itex] and [itex]\partial f/\partial y= 0[/itex] so we have df/dv= y.
 

What is differentiation variable?

Differentiation variable is a mathematical concept used to calculate the rate of change of a function with respect to a specific variable. It is typically denoted as "dx" or "dy" and is an essential aspect of calculus.

How is differentiation variable used in calculus?

In calculus, differentiation variable is used to find the slope of a curve at a specific point. It allows us to calculate the instantaneous rate of change of a function, which is crucial in understanding the behavior of complex systems.

What is the difference between independent and dependent differentiation variables?

The independent differentiation variable is the variable with respect to which the rate of change is calculated, while the dependent differentiation variable is the output variable. In other words, the independent variable determines the value of the dependent variable.

What are the different types of differentiation variables?

There are two main types of differentiation variables: discrete and continuous. Discrete differentiation variables refer to quantities that can only take on specific values, while continuous differentiation variables refer to quantities that can take on any value within a given range.

How can differentiation variable be applied in real-life situations?

Differentiation variable has various applications in real-life situations, such as calculating the velocity of a moving object, determining the growth rate of a population, and predicting stock market trends. It is also used in fields like physics, engineering, and economics to model and analyze complex systems.

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