|Applications of Derivatives→|
We are now ready to define the derivative of a function
We say that ƒ(x) is differentiable at x=a if and only if there exists a real number L such that
L is said to be the derivative of ƒ at a and is denoted by
A function is said to be differentiable on a set A if the derivative exists for each a in A. A function is differentiable if it is differentiable on its entire domain.
Conceptually, finding the derivative means finding the slope of the tangent line to the function. Thus the derivative can be thought of as a linear, or first-order, approximation of the function.
Some properties of the derivative follow immediately from the definition:
If f and g are differentiable, then:
Theorem(Differentiability Implies Continuity)
If f is differentiable at x, it is continuous at x
Since f is differentiable at x, .
Thus , so f is continuous at x.
If f and g are differentiable, then
, since g is continuous at x.
The following theorem is a bit trickier to prove than it seems. We would like to use the following argument:
The problem is that may be zero at points arbitrarily close to x, and therefore would not be continuous at these points. Thus we apply a clever lemma as follows:
Theorem (Caratheodory's Lemma)
We say that is differentiable at if and only if there exists a continuous function that satisfies
()Let be differentiable at and define function such that
It is easy to see that is continuous and that it satisfies the required condition.
() Let be a continuous function satisfying
For all , we have that
As is continuous, , that is,
which implies that is differentiable at
Let be differentiable at
Let be differentiable at
Let the domain of f be a subset of the image of g.
(i) is differentiable at
Caratheodory's Lemma implies that there exist continuous functions such that and
Now, consider the function . Obviously, is continuous.
Also, it satisfies . Hence, by Caratheodory's Lemma, is differentiable at and that
Consider by . What is the derivative of at ?
So, here we see that . Since was an arbtarily choosen point we conculde that
Similar derivative formula's may also be found.
- Find the derivatives of the common functions: Polynomial, Trigonometric, Exponential and Logarithmic.
- Some of the most popular counter examples to illustrate properties of continuity and differentiability are functions involving
(i)Prove that is not continuous at
(ii)Prove that the function is continuous but not differentiable at
(iii)Prove that is differentiable at