93] 
DERIVATIVES AND INTEGRALS 
191 
9. Write down the equations of the tangents at any points of the curves 
y = sinx, y=cosx ; and find where they make a given angle a with the 
axis of x. 
93. We have seen that if <f){x) is not continuous for a value 
of x it cannot possibly have a derivative for that value of x. 
Thus 1 /x 2 cannot have a derivative for x = 0 : nor can any 
function, such as sin(l/#), which is not defined for x = 0 and so is 
necessarily discontinuous for x = 0. Or again the function \x\, 
which is discontinuous for every integral value of x, has no 
derivative for any such value of x. 
Example. Since [.r] is constant between every two integral values of x, 
its derivative is zero for all values of x for which it is defined. Thus the 
derivative of \x\, which we may represent by [#]', is a function equal to zero 
for all values of x save integral values and undefined for integral values. It 
is interesting to note that the functions \x\ and 1 — s !- n are the same 
Sin TVx 
function. 
We saw too in Ex. xxxvm. 7 that the types of discontinuity 
which occur most commonly, when we are dealing with the very 
simplest and most obvious kinds of functions, such as polynomials 
or rational or trigonometrical functions, are associated with an 
equation of the type 
(f> (x) ->■ + oo (or — oo ). 
In all these cases, as in such cases as those considered above, 
there is no derivative for certain special values of x. In fact, as 
was pointed out above, all discontinuities of (f>{x) are also discon 
tinuities of </>' (x). But the converse is not true, as we may easily 
see by considering the special case referred to in § 91, in which the 
graph of <p(x) has a tangent parallel to OY. The most typical 
cases are shown in Fig. 45. In cases (c) and (d) the function is 
o 
Fig. 45. 
X