216 
DERIVATIVES AND INTEGRALS 
[VI 
108. Differentiation of a function of a function. It is 
convenient at this stage to introduce a very important theorem 
which enables us to complete our results relating to the differ 
entiation of the functions which commonly occur in analysis. 
Theorem. If f( x) and F (x) are functions with derivatives 
f (x) and F'(x), then F[f(x)\ has a derivative F'[f(x)} f'(x). 
For let 
/i x ) — V* f(x + h) = y + k, F(y)=z, F(y + k) = z + l. 
Then k and l are functions of h which tend to zero with h, 
and f'(so) and F' (y) or F'{f(x)] are by definition lim (k/h) and 
lim (l/h) respectively. And the derivative of F[f(x)\ is by 
definition 
lim (l/h) = lim (l/k) x lim (k/h) = F'[f(x)]f' (x). 
The following alternative method of proof, depending on the 
Mean Value Theorem, is interesting. It will be noticed that in 
using it we have to assume the continuity and not merely the 
existence of the derivatives of the functions / and F. This is 
of course a theoretical limitation, but not one of any practical 
importance. 
The derivative of F[f(x)\ is by definition 
lim F[f(x + h) -f(x)\' 
h 
But, by the Mean Value Theorem, f(x + h) =f (x) + hf (£), where 
| is a number lying between x and x + h. And 
F (/(*) + hf (f)) = F\f(x)\ + hf (f) F’ (f.), 
where £ is a number lying between f(x) and f(x) + hf(%). 
Hence the derivative of F [f(x)\ is 
lim f'(^)F'(^) = f(x)F / [f(x)], 
since %~>~x and %x-*~f(x) as h^O. 
The most important cases of this theorem are the following : 
(i) If f(x) = ax+b, f (x) = a, and so the derivative of F(ax+b) is 
aF'(ax+b), as result was proved independently in §94, Theorem (8). 
(ii) If f{x)=x m , f'(x)=mx m ~ 1 , and so the derivative of F(x m ) is 
mX ni -1 F' (x m ). 
(hi) If f(x) = ainx, f (x) = cosx, and so the derivative of F(sin a’) is 
cos xF' (sin x). Similarly the derivative of F(cosx) is - sin x F’ (cos x).