Derivatives of Composite Functions (Function of Another Function)

So far, derivatives have only been found for 'simple' functions — $\sin x$, $\log x$, $e^x$ — where the standard-function table gives the answer directly. A composite function is one function applied to the output of another: $\sin\sqrt{x}$ means 'first take $\sqrt{x}$, then take the sine of that'; $\log[\sin(x^2+5)]$ means 'first form $x^2+5$, then take its sine, then take the log of that'. None of the entries in the Class 11 table cover these directly, because the argument is not plain $x$ but a whole expression built from $x$. Before tackling them, it helps to have the full Class 11 standard-derivative list in one place (reproduced below), because every composite-function derivative in this chapter is really just one of these atoms combined with an extra 'multiply by the derivative of the inside' step — which is exactly the chain rule proved in the next section. …