93, 94] 
DERIVATIVES AND INTEGRALS 
193 
In this theorem we of course suppose that f{x) is not zero 
for the particular value of x under consideration. Then 
6m- 
(5) (f>(x) = f(x)/F(x) has a, derivative 
(f)\x) 
f’(x)F(x)-f(x)F'(x) 
[F{x)Y 
This follows at once from (3) and (4). 
(6) (}>(x) = f{x + a) has a derivative 4>'(x) = f\x + a). 
(7) <£ (x) =f{ax) has a derivative ft (x) = af'(ax). 
The proof of the first of these two theorems we leave as an 
exercise to the reader. As regards the second, we have 
<// (x) — a lim [f{ax + ale) — f(ax)]Jah, 
or, putting ax = u and ah = k, 
(})' (x) = a lim {f(u + k) —f{u)]/k = af'(u) = af {ax). 
k^~ 0 
(8) <p(x) —f(ax + b) has a derivative (f>'(x) = af\ax + b). 
This follows at once from (6) and (7). 
Thus if f{x) = sin x,(f) (x) = sin {ax + b), f\x) — cos x, and (f)\x) — a cos {ax+b). 
In particular, if a= 1 and b = ^n, (¡)(x) = cosx and </>'{x) = —sin.r. 
The theorems (6), (7), (8) are capable of simple geometrical interpretations. 
Thus the graph of (p{x)=f{x + a) is obtained by translating that of f{x) 
through a distance a parallel to the axis OX; and the tangent to the curve 
y = (f){x), at the point whose abscissa is x, is parallel to the tangent to the 
curve y=f{x) at the point whose abscissa is x-\-a. 
The graph of 0 {x)=f{ax) may be obtained from that of f{x) by a 
geometrical construction which is probably sufficiently clear from Fig. 46 
(where a = 2). 
H. A. 
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