v + uv ′ (Leibnitz or product rule) ( ) u u v uv v v ′ ′ − ′ = , wherever v ≠ (Quotient rule). The following table gives a list of derivatives of certain standard functions: Table . Whenever we defined derivative, we had put a caution provided the limit exists . Now the natural question is; what if it doesn’t?
The question is quite pertinent and so is its answer. If does not exist, we say that f is not differentiable at c . In other words, we say that a function f is differentiable at a point c in its domain if both – and are finite and equal. A function is said to be differentiable in an interval [ a , b ] if it is differentiable at every point of [ a , b ].
As in case of continuity, at the end points a and b , we take the right hand limit and left hand limit, which are nothing but left hand derivative and right hand derivative of the function at a and b respectively. Similarly, a function is said to be differentiable in an interval ( a , b ) if it is differentiable at every point of ( a , b ). Theorem If a function f is differentiable at a point c , then it is also continuous at that point. Proof Since f is differentiable at c , we have ′ But for x ≠ c , we have f ( x ) – f ( c ) = ( ) .( Therefore lim[ ( ) ( )] c f x .( or lim[ ( )] lim[ ( )] .lim [( )] = f ′ ( c ) .
= or c f x = f ( c ) Hence f is continuous at x = c . Corollary Every differentiable function is continuous. We remark that the converse of the above statement is not true. Indeed we have seen that the function defined by f ( x ) = | x