190 
DERIVATIVES AND INTEGRALS 
[VI 
Example. The reader is no doubt familiar with the formulae 
v = u+ft, s = ut + \ft 2 
for a particle moving in a line with ‘ uniform acceleration /.’ He will easily 
prove from the definition of a derivative that if s=ut + ^ft 2 =(p{t), then 
<p'{t) = u+ft, i.e. cp'(t) = v. The rate of increase of s is v; and similarly the 
rate of increase of v is f 
Examples XLI. 1. If cp{x) is a constant, <p\x)=0. Interpret this 
result geometrically. 
2. If (p{x)=x, (p'{x) = 1. Prove this (a) from the formal definition and 
(6) by geometrical considerations. 
3. If cp{x)=ax+b, <p' (x) = a. Again interpret geometrically. 
4. If (p (x) = x rn , where m is a positive integer, <p'{x)=mx m ~ 1 . 
[For cp' (x) = lim {(a- + h) m - x rn ] /h 
= lim |w.r TO ~ 1 +^^ ^ cc m ~ 2 h +... + h m ~ 1 |. 
The reader should observe that this method cannot be applied to x plq , 
where pjq is a fraction, as we have no means of expressing (x + h) p/q as a 
finite series of powers of h. We shall show later on (§ 98) that the result 
of this example holds for all rational values of m. Meanwhile the reader 
will find it instructive to determine 0' (x) when rn has some special fractional 
value (e.g. £), by means of some special device.] 
5. If <^)(^) = sin.r, <|)'(a?) = cos^; and if 0(^r)=cosA", <£' (#) = — sin x. 
[For example, if c/j (a') = sin x, 
{(p{x+h) — (p (#)}¡h = {2 sin \h cos (x + j>h )}//i 
the limit of which is cos x when 0, since lim cos (x +1-4) = cos x (the cosine 
being a continuous function) and lim {(sin£A)/M} = l (Ex. xxxvn. 14).] 
6. Equations of the tangent and normal to a curve y=cp(x). The 
tangent to the curve at the point {x 0 , y 0 ) is the line through (,r 0 , yf which 
makes an angle \p with OX, where tan \p = cp' (x 0 ). Its equation is therefore 
y-i/o={x-#o)<P'{xo); 
and the equation of the normal (the perpendicular to the tangent at the 
point of contact) is 
(y ~I/o) f fo) +x-x 0 = O. 
We have assumed that the tangent is not parallel to the axis of y. In 
this special case it is obvious that the tangent and normal are x — ,r 0 and 
y=y$ respectively. 
7. Write down the equations of the tangent and normal at any point of 
the parabola x 2 = 4ay. Show that if x 0 =2a/m, y^—a\m 2 is a point of the 
curve, the tangent there is x—my + {alm). 
8. Find the equations of the tangent and normal at any point {m, c 2 /m) 
of the rectangular hyperbola xy = c 2 .