Yes, that makes sense! Thank you for explaining it to me.

In summary, the conversation is discussing the equation $\phi_x \, dx+ \phi_y \, dy=0$ and how it is derived through the chain rule and taking the differential of both sides. The speakers also clarify that this is not technically differentiation, but rather taking the differential.
  • #1
bugatti79
794
1
Hi Folks,

It is been given that differentiation of [tex]\phi(x,y)=0[/tex] is [tex]\phi_{x} dx+ \phi_{y} dy=0[/tex] however I arrive at

[tex]\phi_{x} dx/dy+ \phi_{y} dy/dx=0[/tex] via the chain rule. Where [tex]\phi_{x}=d \phi/dx[/tex] etc

What am I doing wrong?

Thanks
 
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  • #2
What are you differentiating $\phi$ with respect to? Your equation
$$\phi_x \, \frac{dx}{dy}+\phi_y \, \frac{dy}{dx}=0$$
looks schizophrenic - as if you're simultaneously trying to differentiate w.r.t. $x$ and $y$.
 
  • #3
Ackbach said:
What are you differentiating $\phi$ with respect to? Your equation
$$\phi_x \, \frac{dx}{dy}+\phi_y \, \frac{dy}{dx}=0$$
looks schizophrenic - as if you're simultaneously trying to differentiate w.r.t. $x$ and $y$.

Hi, I am referring to eqn 3.2.9 in attached
 

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  • #4
I would say that $\phi_x \, dx+\phi_y \, dy=0$ is really taking the differential of both sides. If you wanted to differentiate w.r.t. $x$, then you would get
$$\phi_x+\phi_y \, \frac{dy}{dx}=0,$$
and if w.r.t. $y$, you'd get
$$\phi_x \, \frac{dx}{dy}+\phi_y=0.$$

Does that answer your question?
 
  • #5
Ackbach said:
I would say that $\phi_x \, dx+\phi_y \, dy=0$ is really taking the differential of both sides. If you wanted to differentiate w.r.t. $x$, then you would get
$$\phi_x+\phi_y \, \frac{dy}{dx}=0,$$
and if w.r.t. $y$, you'd get
$$\phi_x \, \frac{dx}{dy}+\phi_y=0.$$

Does that answer your question?

I think the last 2 equations are coming from the chain rule so yes I can see that.

However, not sure how one arrives at the first equation 3.2.9 by differentiating "both sides"?

Thanks
 
  • #6
bugatti79 said:
I think the last 2 equations are coming from the chain rule so yes I can see that.

However, not sure how one arrives at the first equation 3.2.9 by differentiating "both sides"?

Thanks

It was slightly sloppy wording, in my view. They're really "taking the differential" of both sides. Imagine that $y=f(x)$. If you "took the differential" of both sides, you'd get $dy=f'(x) \, dx$, right? This is not, strictly speaking, differentiation. It's taking the differential. If you were to differentiate, you'd get $y'=f'(x)$.

So, taking the differential of $\phi(x,y)=0$ amounts to doing $\phi_x \, dx+ \phi_y \, dy=0$ in a similar fashion. Does that make sense?
 
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Likes bugatti79

FAQ: Yes, that makes sense! Thank you for explaining it to me.

What is implicit differentiation?

Implicit differentiation is a mathematical technique used to find the derivative of a function that is not explicitly expressed in terms of the independent variable. This technique is particularly useful when the equation of a curve cannot be easily solved for one variable.

How is implicit differentiation different from explicit differentiation?

Explicit differentiation involves finding the derivative of a function that is expressed explicitly in terms of the independent variable. Implicit differentiation, on the other hand, deals with functions that are not expressed explicitly in terms of the independent variable.

When is implicit differentiation used?

Implicit differentiation is used when the equation of a curve cannot be easily solved for one variable. It is also used in cases where the dependent variable is not isolated on one side of the equation, or when there are multiple variables in the equation.

What is the process of implicit differentiation?

The process of implicit differentiation involves treating the dependent variable as a function of the independent variable and using the chain rule to find the derivative. The derivative of the dependent variable is then solved for, and any remaining variables are grouped on one side of the equation.

Why is implicit differentiation important in science?

Implicit differentiation is important in science because it allows us to find the rate of change of a variable that is not explicitly expressed in terms of the independent variable. This technique is used in various fields such as physics, chemistry, and engineering to analyze and model complex systems.

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