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**Prove It** Every function of y is a function of x. So if you wanted to differentiate something like $\displaystyle \displaystyle z= y^2$ (in other words, $\displaystyle \displaystyle z = \left[y(x)\right]^2$ )with respect to x, you need to use the chain rule.

The "inner" function is $\displaystyle \displaystyle y(x)$, and the "outer" function is $\displaystyle \displaystyle y^2$.

So using the chain rule, if $\displaystyle \displaystyle z = y^2$ and we let $\displaystyle \displaystyle u = y \implies z = u^2$, then

$\displaystyle \displaystyle \frac{du}{dx} = \frac{dy}{dx} $ and $\displaystyle \displaystyle \frac{dz}{du} = 2u = 2y$.

Therefore $\displaystyle \displaystyle \frac{dz}{dx}= \frac{dz}{du} \cdot \frac{du}{dx} = 2y\,\frac{dy}{dx}$.

So whenever you want to differentiate a function of y, since y is the inner function, you are always going to get $\displaystyle \displaystyle \frac{dy}{dx}$ as a factor because of the chain rule.