Chain rule problem with partial derivatives

In summary, the conversation discusses solving a chain rule problem with partial derivatives and deriving the second order partial derivative of z with respect to x using the product and chain rule. The correct solution involves taking the derivative of dz/du, which results in a (du/dx)^2 term in the final equation. The conversation also mentions writing the problem in terms of f and g, and requests for a step-by-step solution in LaTeX format.
  • #1
issisoccer10
35
0
[SOLVED] Chain rule problem with partial derivatives

Homework Statement


Suppose that z = f(u) and u = g(x,y). Show that..

[tex]\frac{\partial^{2} z}{\partial x^{2}}[/tex] = [tex]\frac{dz}{du}[/tex] [tex]\frac{\partial^{2} u}{\partial x^{2}}[/tex] + [tex]\frac{d^{2} z}{du^{2}}[/tex] [tex]\frac{(\partial u)^{2}}{(\partial x)^{2}}[/tex]


Homework Equations


[tex]\frac{\partial z}{\partial x}[/tex] = [tex]\frac{dz}{du}[/tex] [tex]\frac{\partial u}{\partial x}[/tex]

based on the chain rule

The Attempt at a Solution


Based on the first order partial derivative above, I would think that using the product rule we can find the second order partial dervative of z w.r.t. x

Using my intuition, I consider [tex]\frac{dz}{du}[/tex] and [tex]\frac{\partial u}{\partial x}[/tex] like different terms and then apply the product rule.

However, I know this isn't correct because I am supposed to show that in the last term of the question equation we have [tex]\frac{\partial u}{\partial x}[/tex] squared, rather than just [tex]\frac{\partial u}{\partial x}[/tex] as I would conclude.

If my attempted solution doesn't make any sense, I'll try to clarify. But it is wrong either way and any help in finding the correct way to get the desired equation would be greatly appreciated.
 
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  • #2
It might be clearer if you write the whole thing in terms of f and g. But you are on the right track, it's product rule and chain rule put together. When you take another d/dx derivative of dz/du you will get the second derivative of z together with yet another du/dx in addition to the one you've already got, giving you a (du/dx)^2. Do you see that, or do I need to tex it?
 
  • #3
oh ok...so

the derivative of [tex]\frac{dz}{du}[/tex] = [tex]\frac{d^{2} z}{du^{2}}[/tex] [tex]\frac{\partial u}{\partial x}[/tex]

because we really are taking the derivative with respect to x.. so basically there was another chain that I didn't see..

is that right?
 
  • #4
issisoccer10 said:
oh ok...so

the derivative of [tex]\frac{dz}{du}[/tex] = [tex]\frac{d^{2} z}{du^{2}}[/tex] [tex]\frac{\partial u}{\partial x}[/tex]

because we really are taking the derivative with respect to x.. so basically there was another chain that I didn't see..

is that right?

I think so. There's a chain you didn't see all right. If you get what you were supposed to then you are doing it right.
 
  • #5
thanks a lot..
 
  • #6


Would anyone tex the solution in a step by step form please?
Thank you in advance
 

1. What is the chain rule in partial derivatives?

The chain rule in partial derivatives is a mathematical rule used to find the derivative of a function composed of two or more functions. It states that the derivative of the composite function is equal to the product of the derivatives of each individual function.

2. How is the chain rule applied in partial derivatives?

To apply the chain rule in partial derivatives, you take the derivative of the outer function and multiply it by the derivative of the inner function. This process is repeated for each function in the composition until you reach the original variable.

3. What is the purpose of using the chain rule in partial derivatives?

The chain rule allows us to find the rate of change of a function with respect to its input variables in a composition of functions. This is important in many applications, such as optimization and curve fitting, where we need to find the optimal value of a function.

4. Are there any special cases for using the chain rule in partial derivatives?

Yes, there are two special cases for using the chain rule in partial derivatives: the chain rule for multivariable functions and the chain rule for implicit functions. In the first case, the function has multiple variables, and in the second case, the function is not explicitly defined in terms of one of its variables.

5. How can I practice and improve my skills in solving chain rule problems with partial derivatives?

The best way to practice and improve your skills in solving chain rule problems with partial derivatives is by doing lots of exercises and problems. You can find many resources online, such as textbooks, practice problems, and video tutorials. Additionally, it is helpful to understand the underlying concepts and not just memorize formulas, as this will help you apply the chain rule in various scenarios.

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