Implicit differentiation: why apply the Chain Rule?

[mcastillo356]

Homework Statement

Calculate $dy/dx$ if $y^2=x$

Relevant Equations

Chain Rule

Hi, PF

$y^2=x$ is not a function, but it is possible to obtain the slope at any point $(x,y)$ of the equation without previously clearing $y^2$. It's enough to differentiate respect to $x$ the two members, treat $y$ like a $x$ differentiable function and make use of the Chain Rule to differentiate $y^2$:

$\dfrac{d}{dx}(y^2)=\dfrac{d}{dx}(x)$

$2y\dfrac{dy}{dx}=1$

$\dfrac{dy}{dx}=\dfrac{1}{2y}$

I can't view $y^2$ like a composite function, instead of just a quadratic expression.

Greetings!

 

[fresh_42]

You have to consider $y^2$ as the composite of two functions, namely $t\stackrel{q}{\longmapsto} t^2$ after $x\stackrel{y}{\longmapsto} y(x)$ which is $(q\circ y)\, : \,x \longmapsto (q\circ y)(x)=q(y(x))=y(x)^2.$

Edit: This leads to an equation of the kind $f(x)=g(x),$ namely $(q\circ y)(x)=\operatorname{id}(x)$ which you differentiated on both sides.

 

[FactChecker]

The examples of implicit differentiation are usually more complicated.
If you are specifically asking about this example, the right side is so simple that it is easy to consider x as a function of y, get the derivative dx/dy = 2y, and take the inverse, dy/dx = 1/(2y) to obtain the slope of y as a function of x.

 

[mcastillo356]

Fine, thanks!