# Total derivative -> partial derivative

• Walkingman
In summary, you can replace a total differential with a partial if the dependency on x is both explicite and implicite.
Walkingman
Under what conditions can you replace a total differential with a partial?

dx/dy -> partial(dx/dy)

in the context of 2 independant variables and multiple dependant variables.

Thanks

let x(u,v),and u,v are functions of y
then
dx/dy=∂x/∂u.∂u/∂y+∂x/∂v.∂v/∂y

but under what conditions can i say

dx/dy = ∂x/∂y ?

Just as dvs77 said: precisely when x depends only on y!

However, that was that really your question? You originally asked "Under what conditions can you replace a total differential with a partial?" A "total differential" is not " a derivative". In other words, not dx/dy. In terms of two independent variables, x and y, the total differential of a function f(x,y) is $$df= \frac{\partial f}{\partial x}dx+ \frac{\partial f}{\partial y}dy$$.

Notice that, in the case you are describing, x and y are not reall "independent".

Thanks

And I was looking for the total derivative, sorry about the mistake in the op.

There's total derivative and there's total differential...Which one are u after...?Halls gave the simplest example of a (total) differential.

The really interesting case is when dependece upon a variable is both explicite & implicite

$$z=z\left(\frac{x}{t},yt^{2},u(t),t\right)$$

and u want to compute

$$\frac{dz}{dt}$$

Daniel.

dvs77 said:
let x(u,v),and u,v are functions of y
then
dx/dy=∂x/∂u.∂u/∂y+∂x/∂v.∂v/∂y

To be more precise with notations this should be

$$\frac{\partial x}{\partial u}\frac{du}{dy}+\frac{\partial x}{\partial v}\frac{dv}{dy}$$

Note that if z=f(x,y) and y=g(x) then d/dx and \partial_x both exist but they are different.

Another interesting case is : z=f(x,z)...(implicit functions)

## 1. What is the difference between total derivative and partial derivative?

The total derivative is the derivative of a function with respect to all of its variables, while the partial derivative is the derivative of a function with respect to one variable while holding all other variables constant.

## 2. How is the total derivative related to the chain rule?

The total derivative can be seen as a generalization of the chain rule, where it takes into account changes in all variables instead of just one.

## 3. Can the total derivative be written in terms of partial derivatives?

Yes, the total derivative can be expressed as a linear combination of the partial derivatives of the function.

## 4. Why is the total derivative used in multivariable calculus?

The total derivative provides a way to calculate the rate of change of a function with respect to all of its variables, making it useful in studying systems with multiple variables and their interactions.

## 5. How does the concept of total derivative extend to higher dimensions?

In higher dimensions, the total derivative becomes the Jacobian matrix, which includes all partial derivatives of a function with respect to all of its variables. This allows for the calculation of higher-order derivatives and more complex systems.

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