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

Dec 3, 2014

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

• $\frac{\Delta}{\Delta x} \left({y}^{2}\right) = \frac{\Delta}{\Delta x} \left(x\right)$

Step 2) To find $\frac{\Delta}{\Delta x} \left({y}^{2}\right)$ we have to use the chain rule because the variables are different.

• Chain rule: $\frac{\Delta}{\Delta x} \left({u}^{n}\right) = \left(n \cdot {u}^{n - 1}\right) \cdot \left(u '\right)$

• Plugging in our problem: $\frac{\Delta}{\Delta x} \left({y}^{2}\right) = \left(2 \cdot y\right) \cdot \frac{\Delta y}{\Delta x}$

Step 3) Find $\frac{\Delta}{\Delta x} \left(x\right)$ with the simple power rule since the variables are the same.

• Power rule: $\frac{\Delta}{\Delta x} \left({x}^{n}\right) = \left(n \cdot {x}^{n - 1}\right)$

• Plugging in our problem: $\frac{\Delta}{\Delta x} \left(x\right) = 1$

Step 4) Plugging in the values found in steps 2 and 3 back into the original equation ( $\frac{\Delta}{\Delta x} \left({y}^{2}\right) = \frac{\Delta}{\Delta x} \left(x\right)$ ) we can finally solve for $\frac{\Delta y}{\Delta x}$.

• $\left(2 \cdot y\right) \cdot \frac{\Delta y}{\Delta x} = 1$

Divide both sides by $2 y$ to get $\frac{\Delta y}{\Delta x}$ by itself

• $\frac{\Delta y}{\Delta x} = \frac{1}{2 \cdot y}$

This is the solution

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