The problem
Every composite you have differentiated so far had an innermost layer that was a plain expression in \(x\), so the last factor of the chain rule block was something like \(2x\) or \(3\). In an implicit equation the variables are mixed, and a term such as \(y^3\) contains no visible \(x\) at all. The layout still applies, but now you have to decide what the innermost layer is.
y is a function of x
An implicit equation such as \(x^2+y^2=25\) does not hand you a formula for \(y\), but near a point on the curve it still defines \(y\) as a differentiable function of \(x\). That is the whole idea behind implicit differentiation: treat \(y\) as shorthand for \(y(x)\), an unknown function of \(x\).
Once you read it that way, \(y^3\) is a composite:
- outer function: \((;)^{3}\), the cubing operation
- inner function: \(y\), the unknown function of \(x\)
The layer inventory has exactly the same shape as the one for \((x^2+1)^3\). Only the inner function has changed.
The block
Apply the established layout: differentiate the outer function with the inner left intact, then multiply by the derivative of the inner function.
\[\frac{d}{dx}\left[y^3\right]=3y^2\cdot\frac{dy}{dx}\]
The first factor, \(3y^2\), is the outer derivative evaluated at the inner function. The second factor, \(\frac{dy}{dx}\), is the innermost layer derivative, because the innermost layer is \(y\) itself and its derivative with respect to \(x\) is \(\frac{dy}{dx}\).
Why the factor is multiplied, not added
The chain rule is a product of one factor per layer. A y-term has the same number of layers as the matching x-term, so it produces the same number of factors. The only difference is that the last factor is written \(\frac{dy}{dx}\) instead of a formula in \(x\).
Common y-terms
| Term | Outer derivative (inner intact) | Innermost factor | Block |
|---|---|---|---|
| \(y^3\) | \(3y^2\) | \(\frac{dy}{dx}\) | \(3y^2\frac{dy}{dx}\) |
| \(\sin y\) | \(\cos y\) | \(\frac{dy}{dx}\) | \(\cos y,\frac{dy}{dx}\) |
| \(e^{y}\) | \(e^{y}\) | \(\frac{dy}{dx}\) | \(e^{y}\frac{dy}{dx}\) |
| \(\ln y\) | \(\frac{1}{y}\) | \(\frac{dy}{dx}\) | \(\frac{1}{y}\frac{dy}{dx}\) |
In every row the \(\frac{dy}{dx}\) factor appears last, exactly where the innermost layer derivative belongs.
Caution
A term that is already a plain function of \(x\), such as \(x^2\), gets no \(\frac{dy}{dx}\) factor. The factor shows up only when the innermost layer is \(y\).