Higher order partial derivatives and the chain rule

• barnflakes
In summary, the conversation is about finding the partial derivative of the function \Phi(u,v) with respect to u, which requires using the chain rule and product rule. The solution provided is the final answer, but it is not obvious how it was obtained. One member suggests starting with the chain rule and substituting in the given functions for x and y. Another member provides a pedantic explanation of the process, while another confirms that they have been able to solve the problem. It is eventually clarified that the missing step in the solution is using the chain rule correctly.
barnflakes

Basically, could somebody please explain to me how I find $${\varphi}_u_u$$, I really don't understand how it's come about. Apparantly I need to use the chain rule again and the product rule but I don't understand how to, if somebody could show me explicitly how to use the product rule in this case I'd be very greatful. The bit under the line is the solution but it's not obvious how they get to that. Thanks for any help!

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• CHAINRULE.GIF
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Anyone?

Here's what I would do:

Take the partial of \phi_{u} with respect to u. Forget about the intermediate step in the solution, I think that required some extra simplification. You care about the end result.

Substitute in for x and y
Pull out of each partial differential any terms that are independent (for example, the partial derivative with respect to u of sin(v)\phi_{x} = sin(v) * partial derivative with respect to u of \phi_{x} )
Then, apply the partial derivatives and substitute back out again for x's and y's. This should get you started, I'm interested to see your work.

Here's how you do it very pedantically:
We have:
$$x(u,v)=e^{u}\cos(v),y(u,v)=e^{u}\sin(v)$$
We now define the function $\Phi(u,v)$ as satisfying identically:
$$\Phi(u,v)=\phi(x(u,v),y(u,v))$$

Thus, for example, we get:
$$\Phi_{u}=\phi_{x}x_{u}+\phi_{y}y_{u}$$
And furthermore:
$$Phi_{uu}=\phi_{xx}x_{u}^{2}+\phi_{xy}x_{u}y_{u}+\phi_{yx}x_{u}y_{u}+\phi_{yy}y_{u}^{2}$$
Got that?

Sorry to dig up an old thread, but it turns out I have been set exactly the same question and after spending hours trying to get my head around it, I still can't get the answer they are looking for. I get the same answer as "arildno" which is no the answer we have been asked to show. Can anyone help again lol.

got it now :D :D :D
$$\phi_{uu}=(\frac{\partial}{\partial x}(x\phi_{x}+y\phi_{y}))x_{u}+(\frac{\partial}{\partial y}(x\phi_{x}+y\phi_{y}))y_{u}$$

Finally worked out how to use the chain rule correctly. This is the middle step that is missing.

1. What is a higher order partial derivative?

A higher order partial derivative is a derivative of a function with respect to one or more of its variables, taken multiple times. It represents how the rate of change of the function changes as the independent variables change.

2. How is the chain rule used to find higher order partial derivatives?

The chain rule is used to find higher order partial derivatives by breaking down the function into smaller parts, finding the derivatives of those parts, and then combining them using the chain rule formula.

3. What is the notation for a higher order partial derivative?

The notation for a higher order partial derivative is similar to that of a regular derivative, with the addition of subscripts to indicate which variables the derivative is taken with respect to. For example, a second order partial derivative of a function f with respect to x and y would be written as fxx or fyy.

4. Can a higher order partial derivative be negative?

Yes, a higher order partial derivative can be negative if the function is decreasing in one direction and increasing in another direction. This indicates a change in the rate of change in the function.

5. What is the significance of higher order partial derivatives in real-world applications?

Higher order partial derivatives are essential in understanding the behavior of functions in multi-variable systems. They are used in fields such as physics, economics, and engineering to model complex systems and make predictions about their behavior.

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