I believe I understand the chain rule better from a few tutorials as the following:
$$\frac{d}{dx}(f(g(x)) ) = \frac{\partial f}{\partial g}\frac{\partial g}{\partial x}$$
But how would you represent equations that require the chain rule with leibniz notation for higher order derivatives. For example would the second derivative of a function requiring the chain rule be represented as follows:
$$\frac{d^2}{dx^2}(f(g(x)) ) = \frac{\partial^2 f}{\partial g}\frac{\partial g}{\partial x^2}$$
Or will the product rule be required somewhere? Any information would be appreciated
Best Answer
Please do not use notation like this. If $f=f(u)$ and $u=g(x)$, please write $$\frac d{dx}f(g(x)) = \frac{df}{du}\Big|_{u=g(x)}\cdot \frac{dg}{dx}.$$ You then need to use the product rule and chain rule to take the derivative again. In particular, $$\frac d{dx}\left(\frac{df}{du}\Big|_{u=g(x)}\right) = \frac{d^2f}{du^2}\Big|_{u=g(x)}\cdot\frac{dg}{dx}.$$ Now you finish up with the product rule.