# Implicit differentiation: why apply the Chain Rule?

• mcastillo356
In summary, the conversation discusses how to obtain the slope at any point on the equation y^2=x without previously clearing y^2. It involves differentiating the equation and treating y as a differentiable function. This can be done by considering y^2 as a composite function and using the Chain Rule. However, the example given is simple enough to just consider x as a function of y and take the inverse to obtain the slope of y as a function of x.

#### mcastillo356

Gold Member
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!

• Delta2
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.

• • mcastillo356, FactChecker and Delta2
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!