Let us now deduce the condition(s) on the derivative of a function which determines whether is increasing/decreasing on a given interval. We are assuming that is everywhere differentiable.
The function on the left side, , is a strictly increasing function. Notice that the slope of the tangent drawn at any point on this curve is always positive. Hence, a sufficient condition for to be strictly increasing on a given domain is
Later on, we will see that this is not a necessary condition for a function to be strictly increasing.
In Fig, the function on the right side, is not strictly increasing though it is increasing. Notice that or is never negative. Hence, a sufficient condition for to be increasing on a given domain is
Now consider and in Fig
Extending the previous case, we get the conditions for a (strictly) decreasing function:
Strictly decreasing:
 
Decreasing:

The remarks made for the increasing case hold true here also.
Before concluding this section, here are some other facts worth paying attention to:
(a) If is strictly increasing, then is also strictly increasing. Similarly, if is strictly decreasing then is also strictly decreasing.
(b) If and have the same monotonicity (both increasing or decreasing) on , then and are monotonically increasing on .
(c) If and have opposite monotonicity on , then and are strictly decreasing on
(d) The inverse of a continuous function is continuous
(e) If except for a finite (or an infinitely countable) number of points where is still strictly increasing on . This is why we said earlier that is not a necessary condition for strict increase. For example, in a later example we will consider the graph of the function . We will see that is not always positive ; even then, increases strictly, because the points at which are countable
(f) Similarly if except for a finite (or an infinitely countable) number of points where is still strictly decreasing on