Implicitly differentiating PDE (multivariable calculus)

In summary, to find the value of dz/dx at the point (1,1,1), you can use the product rule and the chain rule to solve the equation xy+z3x-2yz=0. First, implicitly differentiate z^3x to get 3z^2x*dz/dx + z^3. Then, use the product rule to differentiate the remaining terms, resulting in the equation y + 3x(dz/dx)2 + z3 - 2y(dz/dx) = 0. Finally, solve for dz/dx to find the value at the given point.
  • #1
Legion81
69
0
The problem:
Find the value of dz/dx at the point (1,1,1) if the equation xy+z3x-2yz=0 defines z as a function of the two independent variables x and y and the partial derivative exists.

I don't know how to approach the z3x part. I thought you would use the product rule and get 3(dz/dx)2x + z3. But if that is right, the final equation looks something like

y + 3x(dz/dx)2 + z3 - 2y(dz/dx) = 0

And I don't think that is right. The only way I know to solve that would be with the quadratic equation and that gives a complex value. Am I forgeting the chain rule somewhere or just way off on approaching this problem?
 
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  • #2
I have done very little multivariable so this could easily be wrong, but if you implicitly differentiated z^3x shouldn't you get 3z^2x*dz/dx + z^3?
 
  • #3
Frillth said:
I have done very little multivariable so this could easily be wrong, but if you implicitly differentiated z^3x shouldn't you get 3z^2x*dz/dx + z^3?

That's right because you have the 3z2x(dz/dx) + z3(dx/dx). I don't know what I was thinking... Thank You!
 
  • #4
You didn't apply the chain rule, remember df(u)/dx = f(u)'*du/dx
 

FAQ: Implicitly differentiating PDE (multivariable calculus)

What is implicit differentiation in multivariable calculus?

Implicit differentiation is a method used to find the derivatives of functions that are not explicitly written in terms of a single independent variable. In multivariable calculus, this method is applied to partial differential equations (PDEs) where the dependent variable is a function of multiple independent variables.

How is implicit differentiation different from explicit differentiation?

In explicit differentiation, the dependent variable is explicitly written as a function of the independent variable and the derivatives are found by using the basic differentiation rules. On the other hand, implicit differentiation is used when the dependent variable is not explicitly written in terms of the independent variable, and the derivatives are found by treating the dependent variable as an implicit function of the independent variables.

What are the steps involved in implicitly differentiating a PDE?

The following are the general steps involved in implicitly differentiating a PDE:
1. Identify the dependent variable and the independent variables in the PDE.
2. Take the derivative of both sides of the equation with respect to each independent variable, treating the dependent variable as an implicit function.
3. Simplify the resulting equations using the chain rule and other differentiation rules.
4. Solve the resulting system of equations for the derivatives of the dependent variable.

What are the applications of implicit differentiation in PDEs?

Implicit differentiation is commonly used in physics and engineering to model complex systems and phenomena. It can be used to solve PDEs in various fields such as fluid dynamics, electromagnetism, and heat transfer. It also has applications in economics, biology, and other sciences.

What are the limitations of implicit differentiation in PDEs?

Implicit differentiation can only be applied to PDEs where the dependent variable is a differentiable function of the independent variables. It also becomes more complex when dealing with higher-order derivatives. Additionally, it may not always produce an explicit solution and may require further analysis or numerical methods to find a solution.

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