Question on Implicit function

Question

Question - Show that $$\frac{\partial f}{\partial y}~\frac{\partial Ⲫ }{\partial z}~\frac{dz}{dx}=~\frac{\partial f }{\partial x}~\frac{\partial Ⲫ }{\partial y}~,$$where $~f(x,y)=0;~~ Ⲫ (z,y)=0~.$

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Sol. In the function $~f(x,y)=0~,~~ y~$ is an implicit function of $~x~$

$$\frac{dy}{dx}=\frac{\frac{\partial f }{\partial x }}{\frac{\partial f }{\partial y}}$$

And the solution goes on...

My query is “In the function $~f(x,y)=0~, ~~y~$ is an implicit function of $~x~$” , how we know that $~y~$ is an implicit function of $~x~$?

Answer 1

Explicit Function: A function in which the dependent variable, and independent variable are separated on opposite sides of the equality are known as explicit function.

i.e., If a function is written (or can be written) as $~y=f(x)~$, then it is explicit.

e.g., Take $~x^2+y^2-1=0~$, solving for $~y~$ gives an explicit solution: $~y=\pm \sqrt{1-x^2}~$.

Implicit Function: A function in which the dependent variable, and independent variable(s) are not separated (isolated) on opposite sides of the equality are known as implicit function.

i.e., If it only has the form $~f(x,y)=0~$, then it is implicit.

e.g., Take $~x^2 + xy~ – y^2 = 1~$, then $~f(x,y)=x^2 + xy~ – y^2-1=0~$.

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I think my above discussion will clear your doubt. You can also see here.

Answer 2

We have to use the Implicit Function Theorem.

We consider the curve defined by $f(x, y)=0$. Now, pick two points $x_0$ and $y_0$ such that $f(x_0,y_0)=0$. If it holds $$ \left. \frac{\partial f}{\partial y} \right|_{(x_0, y_0)} \neq 0, $$ then there exists a differentiable function $y(x)$ such that, for $x$ close to $x_0$, $$ f(x, y(x)) = 0. $$

In particular, $$ \frac{\partial y}{\partial x} = \frac{ \frac{\partial f}{\partial x} }{ \frac{\partial f}{\partial y} }. $$

I hope this helps.