Calculus Tutorials - Derivatives (III)
Notation: The derivative ) of a function ) is
also denoted as
)
This notation comes from the fact that when you compute the
derivative, you compute
 = \displaystyle\lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0})
and the term -f(x_0)) is usually referred as  , and
the term  is referred as  . So, sometimes, in some
books (specially Physics books) you're going to find the definition
 = \displaystyle\lim_{\Delta x\to 0} \frac{\Delta f}{\Delta x})
Theorems to calculate Derivatives
Now it is the time to introduce the heavy artillery. In practice,
you won't be computing the limit
 = \displaystyle\lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0})
very often. It is very important to know how to do it that way, but
most of the times it won't be necessary.
Example: Compute the derivative of the function
=x^3+x^2) .
Solution: What do we do here, do we apply the
limit to compute the derivative?? Well, your line of reasoning
should be the following: The function  = x^3+x^2) corresponds
to the sum of  and  . The intuition is that if I could
compute the derivative of each term separately, then I
could simplify the calculation.
In other words, if I knew what is the derivative of , and if I
also knew what is the derivative of , then I should know what
is the derivative of  .....
 In fact, you do. We have the following theorem:
Theorem: The Derivative of the Sum
Assume that and ) are differentiable at 
(that means that the derivative exists at that point). Then, we have
that
+g(x)) = \frac{df}{dx}(x) + \frac{dg}{dx}(x) )
In other words, the derivative of the sum is the sum of the derivatives (These are not empty words, they really describe the result accurately).
This is usually referred as the Linearity Property of the derivative
Now we show a result that will help us to compute a lot of
derivatives:
Theorem: The following holds true for all  :
Proof: We won't do many proofs, to save you from boredom, but let's just do this one to get a feel of it. By definition
 = \displaystyle\lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}=\displaystyle\lim_{x\to x_0} \frac{x^n- x_0^n}{x-x_0})
(x^{n-1}+x^{n-2}x_0 + ...+ xx_0^{n-2} +x_0^{n-1})}{x-x_0} = \displaystyle\lim_{x\to x_0}(x^n+x^{n-1}x_0 + ...+ xx_0^n +x_0^n) )
So, let's come back to the problem of finding the derivative of . Using the Linearity of derivatives we find that
 = \frac{d}{dx} x^3 + \frac{d}{dx}x^2 = 3x^2 + 2x)
Let's recall that  , so applying that to the case  and  respectively we get the previous result.
The Linearity property can be written in a more general way:
Theorem: Assume that and are differentiable at and  and  are constants. Then
+ bg(x)) = a\frac{df}{dx}(x) + b\frac{dg}{dx}(x) )
Below we show an example of how to apply this result:
Example: Compute the derivative of the function =3 x^3+2 x^{1/2}) .
Solution: Using Linearity, we get that
 = 3\frac{d}{dx} x^3 + 2 \frac{d}{dx}x^{1/2} = 3(3x^2) + 2\left(\frac{1}{2} x^{-1/2}\right))
|
|
