Implicit Differentiation

In summary, the answer key differentiated the equation by using the chain rule. They solved for y' by differentiating (x/y) = ln(x-y). Differentiating (x/y) = ln(x-y) is more difficult, but still possible.
  • #1

Homework Statement


Find y' in
e^(x/y)=x-y

2. The attempt at a solution
I tried to differentiate it by changing it so that there would be a natural log (as seen in my attachment). However the end result is not the same as the answer key.

How the answer key did it was they used the chain rule on e^(x/y). So...

e^(x/y)*(x/y)'=1-y'

And then they solve for y'.


Why can I not change the e^(x/y) to the logarithmic notation?
 

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  • #2
JustinLiang said:

Homework Statement


Find y' in
e^(x/y)=x-y

2. The attempt at a solution
I tried to differentiate it by changing it so that there would be a natural log (as seen in my attachment). However the end result is not the same as the answer key.

How the answer key did it was they used the chain rule on e^(x/y). So...

e^(x/y)*(x/y)'=1-y'

And then they solve for y'.

Why can I not change the e^(x/y) to the logarithmic notation?
Without seeing exactly what you did, there is no way to say whether or not your method is correct.

Is it any easier to find y' by differentiating (x/y) = ln(x-y) ?
 
  • #3
SammyS said:
Without seeing exactly what you did, there is no way to say whether or not your method is correct.

Is it any easier to find y' by differentiating (x/y) = ln(x-y) ?

Oops, I forgot to attach it. It is harder but I was just doing it for practice, it seems like I was unable to get the correct answer... Do you know why? I attached the photo.
 
  • #4
JustinLiang said:
Oops, I forgot to attach it. It is harder but I was just doing it for practice, it seems like I was unable to get the correct answer... Do you know why? I attached the photo.
It does look difficult to get the answer into the same form once you get rid of the exponential. Taking the derivative then gets rid of the logarithm.

You do have a mistake in your work. [itex]\displaystyle \frac{d}{dx}\ln(x-y)=\frac{1-y'}{x-y}\,.[/itex]

Added in Edit:

Actually, it's not that difficult to compare the results. Take the book answer and replace [itex]\displaystyle e^{x/y}[/itex] with [itex]x-y\,.[/itex]
 
Last edited:
  • #5
SammyS said:
It does look difficult to get the answer into the same form once you get rid of the exponential. Taking the derivative then gets rid of the logarithm.

You do have a mistake in your work. [itex]\displaystyle \frac{d}{dx}\ln(x-y)=\frac{1-y'}{x-y}\,.[/itex]

Added in Edit:

Actually, it's not that difficult to compare the results. Take the book answer and replace [itex]\displaystyle e^{x/y}[/itex] with [itex]x-y\,.[/itex]

Wow... I need some sleep haha. Thanks.
 

What is implicit differentiation?

Implicit differentiation is a method used in calculus to find the derivative of a function that is not expressed explicitly in terms of one variable. It is commonly used when a function is defined implicitly as an equation involving multiple variables.

How is implicit differentiation different from explicit differentiation?

The main difference between implicit differentiation and explicit differentiation is that implicit differentiation involves finding the derivative of a function that is not expressed explicitly in terms of one variable, whereas explicit differentiation involves finding the derivative of a function that is expressed explicitly in terms of one variable.

What is the process of implicit differentiation?

The process of implicit differentiation involves taking the derivative of both sides of an equation with respect to a specific variable. This allows us to find the derivative of the function with respect to that variable without having to solve for it explicitly.

What are the advantages of using implicit differentiation?

One advantage of using implicit differentiation is that it allows us to find the derivative of a function without having to solve for it explicitly. This can be useful when the function is too complex to solve explicitly, or when the function is defined implicitly.

What are some common applications of implicit differentiation?

Implicit differentiation is commonly used in physics and engineering to solve problems involving rates of change, such as velocity and acceleration. It is also used in economics and finance to analyze the relationships between variables in more complex models.

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