Chain Rule & PDES: Solving ∂z/∂u

In summary, the conversation discusses a question about the chain rule and partial differential equations. The appropriate version of the chain rule for the derivative is discussed and a dependence chart is mentioned. The correct formula for the derivative is provided and a diagram is suggested to help understand the concept.
  • #1
Kork
33
0
Im new on the forum, so I hope you guys will have some patience with me :-)

I have a question about the chain rule and partial differential equations that I can't solve, it's:

Write the appropriate version of the chain rule for the derivative:

∂z/∂u if z=g(x,y), where y=f(x) and x=h(u,v)

I have tried to do a dependence chart of it, but it's just not working for me.

Thank you very much.
 
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  • #2
∂z/∂u=(∂z/∂x)(∂x/∂u)+(∂z/∂y)(dy/dx)(∂x/∂u)

Draw a diagram. It is pretty straightforward.
 

FAQ: Chain Rule & PDES: Solving ∂z/∂u

What is the chain rule and how is it used in solving ∂z/∂u?

The chain rule is a mathematical rule that allows us to find the derivative of a composite function. In the case of solving ∂z/∂u, it is used to find the partial derivative of a function with respect to a variable that is indirectly dependent on another variable.

Why is the chain rule important in partial differential equations (PDEs)?

In many cases, PDEs involve functions that are composed of multiple variables. The chain rule allows us to break down these complex functions and find their partial derivatives, making it an essential tool in solving PDEs.

What are the steps for applying the chain rule to solve ∂z/∂u?

The steps for applying the chain rule to solve ∂z/∂u are as follows:

  1. Identify the function z that is expressed in terms of u.
  2. Identify the intermediate variable involved in the composite function.
  3. Take the partial derivative of z with respect to the intermediate variable.
  4. Take the partial derivative of the intermediate variable with respect to u.
  5. Multiply the two derivatives together to get the final result: ∂z/∂u.

Can the chain rule be applied to functions with more than two variables?

Yes, the chain rule can be applied to functions with any number of variables. The process remains the same, but the number of intermediate variables may increase depending on the complexity of the function.

Are there any limitations to using the chain rule in solving PDEs?

The chain rule can only be used for functions that are differentiable. If a function is not continuous or has a discontinuity, the chain rule may not be applicable. Additionally, the chain rule may become more complicated when dealing with higher-order derivatives in PDEs.

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