Special vs. General Case in Basic Algebraic Notation

Question

From Rogawski, Jon (2011-04-01). Single Variable Calculus (Page 343). W. H. Freeman. :

To compute the derivative of a function of the form e^g(x), write e^g(x) as a composite e^g(x) = f(g(x) ), where f(u) = e^u, and apply the Chain Rule:

[ f(g(x) ) ]' = f'(g(x) ) * g'(x) = e^g(x) * g'(x) [ because f'(x) = e^x ]

A special case is ( e^kx+b )' = ke^kx+b, where k and b are constants.

Is f(g(x) ) = e^kx+b really a special case? Isn't this the general case?

Answer 1

No, the case $\mathrm{g}(x) = kx+b$, i.e. where $\mathrm{g}$ is a linear function, is really quite special.

Even amongst all possible polynomials $a_nx^n + a_{n-1}x^{n-1}+\cdots + a_1x + a_0$, the linear functions are a special example. When you include trigonometric function, logarithms, rational functions, hyperbolic functions and other non-elementary functions, you see that $\mathrm{g}(x)=kx+b$ is special.

What about, for example,

$$\mathrm{g}(x) = \sin(\cos(\ln(\sin(\cos(\mathrm{e}^x))))$$

Answer 2

The linear function in the exponent is an easy example. In general you could have a much more complicated function for $g(x)$.

For example if $f(x) = e^{\textrm{sin}(x)}$ then

$f'(x) = \textrm{cos}(x)e^{\textrm{sin}(x)}$

In every case the derivative of $f(g(x))$ is just $f(g(x))*g'(x)$.