# Partial derivitives chain rule proof

• ProPatto16
In summary, to show that d2u/dx2+d2u/dy2 = e-2s[d2u/ds2+d2u/dt2, you can use the chain rule and the fact that x= e^s cos(t) and y= e^s sin(t) to find the second partial derivatives. The remaining terms can be simplified to show that the two sides are equal.
ProPatto16

## Homework Statement

If u=f(x,y) where x=escost and y=essint

show that d2u/dx2+d2u/dy2 = e-2s[d2u/ds2+d2u/dt2

## The Attempt at a Solution

i have no idea!

question though, do the partial derivitives have to be solved and expanded then just show that one side equals the other or can it be proven by manipulating the terms as they are?

the only relevant chain rule is this one

du/ds=du/dx*dx/ds+du/dy*dy/ds and du/dt=du/dx*dx/dt+du/dy*dy/dt

this gives the first step of the partial derivitives to some of the question. how would i go about finding the second partial derivitive?

any help would be great!

You find the second partial derivative by doing it again!

For example, $x= e^s cos(t)$ and $y= e^s sin(t)$ so
$$\frac{\partial f}{\partial s}= \frac{\partial f}{\partial x}(e^s cos(t)+ \frac{\partial f}{\partial y}(e^s sin(t)$$

Then
$$\frac{\partial^2 f}{\partial s^2}= \frac{\partial}{\partial s}\left(\frac{\partial f}{\partial x}(e^s cos(t)+ \frac{\partial f}{\partial y}(e^s sin(t)\right)$$
$$= \frac{\partial}{\partial s}\left(\frac{\partial f}{\partial x}\right)(e^s cos(t))+$$$$\frac{\partial f}{\partial x}\left(\frac{\partial}{\partial s}\right)\left(\frac{\partial}{\partial s}\right)\left(e^s cos(t)\right)+$$$$\frac{\partial f}{\partial y}(e^s sin(t))+ \frac{\partial}{\partial s}\left(e^s sin(t)\right)$$
$$= \frac{\partial^2 f}{\partial x^2}(e^s cos(t))^2+\frac{\partial f}{\partial x}(e^s cos(t)) \frac{\partial^2 f}{\partial x^2}(e^s sin(t))^2+ \frac{\partial f}{\partial x}(e^s sin(t))$$

Last edited by a moderator:
in the last line... should there be a + in the middle there?

Last edited:
assuming there is meant to be a plus in there... and continuing on...

that last line simplifies to:

= d2f/dx2(e2scos2t+e2ssin2t) + df/dx(escost+essint)

= d2f/dx2[e2s(cos2t+sin2t)] + df/dx(escost+essint)

now i can see that cos2t+sin2t = 1 so first term becomes d2f/dx2[e2s]

then when you did it with d2f/dy2 you would get d2f/dy2[e2s] for the first term then a similar second second, then i can see how the proof would work out

but i can't get rid of the second terms df/dx(escost+essint) and the df/dy one ?

hold up...

f in terms of x and y would just be f=x+y right? without making the substitutions for the equations of x and y... therefore df/dx would be zero? then it all works out!

## 1. What is the chain rule for partial derivatives?

The chain rule for partial derivatives is a method for finding the derivative of a composite function. It states that the derivative of a composition of two functions is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

## 2. How is the chain rule applied when finding partial derivatives?

To apply the chain rule when finding partial derivatives, we treat each variable as if it were the only variable in the function and differentiate it with respect to that variable. We then multiply the resulting derivatives by the chain rule.

## 3. Can the chain rule be used for higher order partial derivatives?

Yes, the chain rule can be used for higher order partial derivatives. To find the second or higher order partial derivative, we simply apply the chain rule repeatedly, taking the derivative of the outer function and then the inner function until we have the desired derivative.

## 4. Are there any exceptions to the chain rule for partial derivatives?

There are some exceptions to the chain rule for partial derivatives, such as when the inner function is not differentiable or when the functions are not continuous. In these cases, the chain rule may not apply or may need to be modified.

## 5. How is the chain rule used in real-world applications?

The chain rule is used in various fields of science, such as physics, engineering, and economics, to model and analyze complex systems. It allows us to calculate the rate of change of a dependent variable with respect to multiple independent variables, which is crucial in understanding and predicting the behavior of these systems.

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