What is the derivative of `x=y^2`?

We can solve this problem in a few steps using Implicit Differentiation.
Step 1) Take the derivative of both sides with respect to x.

• `(Delta)/(Deltax)(y^2)=(Delta)/(Deltax)(x)`

Step 2) To find `(Delta)/(Deltax)(y^2)` we have to use the chain rule because the variables are different.

• Chain rule: `(Delta)/(Deltax)(u^n)= (n*u^(n-1))*(u')`

• Plugging in our problem: `(Delta)/(Deltax)(y^2)=(2*y)*(Deltay)/(Deltax)`

Step 3) Find `(Delta)/(Deltax)(x)` with the simple power rule since the variables are the same.

• Power rule: `(Delta)/(Deltax)(x^n)= (n*x^(n-1))`

• Plugging in our problem: `(Delta)/(Deltax)(x)=1`

Step 4) Plugging in the values found in steps 2 and 3 back into the original equation ( `(Delta)/(Deltax)(y^2)=(Delta)/(Deltax)(x)` ) we can finally solve for `(Deltay)/(Deltax)`.

• `(2*y)*(Deltay)/(Deltax)=1`

Divide both sides by `2y` to get `(Deltay)/(Deltax)` by itself

• `(Deltay)/(Deltax)=1/(2*y)`

This is the solution

Notice: the chain rule and power rule are very similar, the only differences are:
-chain rule: `u!=x` "variables are different" and
-power rule: `x=x` "variables are the same"