2.3 Basic Derivative Rules
We’ll next forge some tools so we can leave the limit definition of derivative behind (we will no longer need those crude stone implements!).
First a little notation… Assuming that y = f(x), recall the notations we have for the derivative:
If we rewrite that first “fraction” slightly, we get:
This actually gives us an entirely new concept, that we’ll be using quite a bit henceforth – the concept of a differential operator. Just as a function inputs numbers and outputs numbers, and operator inputs functions and outputs functions. So an operator is a kind of function of functions. We’ve seen how the derivative process starts with one function, but ends up with a quite different function. Think of as an operator, that’s waiting patiently to eventually take a derivative (with respect to variable x) of whatever is to it’s immediate right:
In particular, this notation is useful in stating the following properties of the derivative.
Basic Derivative Properties
Assume c and n to be real constants. Then the following theorems hold true:
For proofs of these theorems, click here.
Properties  and  together mean that d/dx is a linear operator.
Equipped with these rules, we can compute the derivatives of many types of functions very easily, and without having to take any limits. In particular, we can easily deal with the derivatives of polynomial functions.
Example 1: f(x) = x3 + 3x 2 – 5x + 1. Find f ’(x).
That was MUCH easier than using the limit definition. The above shows which properties justified each step; in practice you can use them all together, and so calculate the derivative in just one step.
Example 2: g(x) = x5 + 2x 3 + 7x + 3. Find g’(x).
Here are the justifications for this one step…
A:  B: , C: , D: 
Notice how  and  together allow us to multiply the exponent times the coefficient, subtracting 1 from the exponent.
Example 3: h(x) = x100 – 5x50 + 9x10 – 12. Find h’(x).
The power rule works for all real numbers x (well, it even works for complex-valued x, but that’s another course!).
Example 4: (power rule)
Example 5: (power rule)
Example 6: (power rule)
Compute the derivative of the function .
But the derivative of a sum is the sum of the derivatives, so this
and the derivative of a constant times a function is the constant times the derivative of the function (and the derivative of a constant is 0), so
Lastly, apply the power rule and simplify:
Of course, once you get used to doing these kinds of derivatives, you'll do this all in one step!
Example 8: f(x) = 2/x5 . Find f ’(x).
Compute the derivative of the function
The key to this
problem it to first write the function using exponent notation:
Now just take the derivative, using the following three derivative rules:
* the derivative of a sum is the sum of the derivatives;
* the derivative of a constant times a function is
the constant times the derivative of the function;
* the power rule for derivatives
It’s a good
rule of thumb to write down a solution in the same form in which the original
problem was expressed, so this becomes:
2.3 Exercises Do all of them! :-)