Practical:Chain rule for differentiation

This article considers practical aspects of the chain rule for differentiation: how is this rule used in actual computations?

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Statement to remember

The statement of the chain rule for differentiation that we will be using is:

${\frac {d}{dx}}[f(g(x))]=f'(g(x))g'(x)$

where $f'(u)={\frac {d(f(u))}{du}}$ and $g'(x)={\frac {d(g(x))}{dx}}$.

NOTE: As a matter of convention, and to reduce confusion, we use a different variable ($u$ in this case) for the generic input to $f$ compared to the variable ($x$ in this case) that we use for the generic input to $g$.

Procedure to apply the chain rule for differentiation

The chain rule for differentiation is useful as a technique for differentiating functions that are expressed in the form of composites of simpler functions.

Most explicit procedure

The explicit procedure is outlined below:

1. Identify the two functions whose composite is the given function. In other words, explicitly decompose the function as a composite of two functions. We will here call the functions $f$ and $g$, though you may choose to give them different names.
2. Calculate the derivatives of $f$ and $g$ separately, on the side.
3. Plug into the chain rule formula the expressions for the functions and their derivatives.
4. Simplify the expression thus obtained (this is optional in general, though it may be required in some contexts).