How can I solve this problem using implicit differentiation?

In summary, Mark 44 was trying to find dw/dy(1/(w^2+x^2)+1/(w^2+y^2)) and had trouble because w^2+x^2 does not involve y and the partial derivatives with respect to y are not zero. After getting help from other users, he realized that he needed to differentiate dw/dy(1/(w^2+x^2)+1/(w^2+y^2)) with respect to y.
  • #1
Sebastian B
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I unfortunately keep on getting the wrong answer to this problem.

I am supposed to find: dw/dy(1/(w^2+x^2)+1/(w^2+y^2))

I attached a picture of how I tried to solve it. Help would be much appreciated.
 

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  • #2
Sebastian B said:
I unfortunately keep on getting the wrong answer to this problem.

I am supposed to find: dw/dy(1/(w^2+x^2)+1/(w^2+y^2))
No. What you wrote above doesn't mean what you think.
I believe what you are supposed to do is use implicit differentiation to find ##\frac{\partial w}{\partial y}##, although that is not clear from what you wrote on the first line. On the third line you have a mistake. Since ##\frac{1}{w^2 + x^2}## does not involve y, its partial derivative with respect to y is zero.
Sebastian B said:
I attached a picture of how I tried to solve it. Help would be much appreciated.
 
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Likes Sebastian B
  • #3
That 2y should also be part of the multiplication. Double check your application of the chain rule on (w^2+y^2).
 
  • #4
Sebastian B said:
I unfortunately keep on getting the wrong answer to this problem.

I am supposed to find: dw/dy(1/(w^2+x^2)+1/(w^2+y^2))
You mean that you need to find ∂w/∂y given that (1/(w2+x2)+1/(w2+y2)) = 1. Right ?

I attached a picture of how I tried to solve it. Help would be much appreciated.
Hello Sebastian B. Welcome to PF !

In your third equation, you need to have parentheses around ##\displaystyle \ \left( 2w\frac{\partial w}{\partial y}+2y \right) \ ## .

After that the algebra is messed up.

Don't forget; the right side of the equation is zero at that point.
 
  • #5
Mark44 said:
No. What you wrote above doesn't mean what you think.
I believe what you are supposed to do is use implicit differentiation to find ##\frac{\partial w}{\partial y}##, although that is not clear from what you wrote on the first line. On the third line you have a mistake. Since ##\frac{1}{w^2 + x^2}## does not involve y, its partial derivative with respect to y is zero.

Mark 44: Thank you for your feedback. I realized that I wasn't clear enough with stating my problem, but you figured out what I meant. However,
\[\delta\]w/\[\delta\]y (w^2+x^2) does not equal zero, because w could be dependent on y
 
  • #6
Ruber and Sammys thanks a lot for the feedback, it helped a lot! I finally figured it out and got the correct answer thanks to you guys. What a dumb error that was. Thanks again!
 

What is implicit differentiation?

Implicit differentiation is a mathematical technique used to find the derivative of an equation that cannot be easily solved for one variable. It allows us to find the rate of change of a function with respect to one of its variables, even if the function is not explicitly written in terms of that variable.

When is implicit differentiation used?

Implicit differentiation is used when the equation of a curve or surface cannot be easily solved for one of its variables. This commonly occurs when the equation contains both the dependent and independent variables together, making it difficult to isolate the dependent variable.

How is implicit differentiation performed?

To perform implicit differentiation, we treat the dependent variable as a function of the independent variable and use the chain rule to differentiate it with respect to the independent variable. Then, we solve for the derivative by collecting terms with the dependent variable on one side and all other terms on the other side.

What are the advantages of using implicit differentiation?

One advantage of implicit differentiation is that it can be used to find the derivative of a function without needing to explicitly solve for one of its variables. This can be helpful when the equation is difficult or impossible to solve explicitly. It also allows us to find the derivative of implicit functions, which cannot be expressed in terms of a single equation.

Are there any limitations to implicit differentiation?

Implicit differentiation can only be used to find the derivative of a function with respect to one variable. If we want to find the derivative with respect to multiple variables, we would need to use partial differentiation. Additionally, implicit differentiation may not always yield an explicit formula for the derivative, so it may not be useful in all situations.

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