, we have to compute the limitThe examples of derivative calculations we saw last week were somewhat depressing, because it seemed that each different example required yet another special trick to solve. Will we need to perform astounding feats of algebra each time we compute a derivative? Fortunately, there is an easier way. We are going to indulge ourselves in an approach that is quite common in mathematics:
• Solve certain ‘key’problems.
• Discover ways to combine those problems into larger problems.
What are the ‘key’ problems we need to solve? To start with, here is one of the more important ‘key’ problems.
The derivative of xn
To compute the derivative of , we have to compute the limit
The strategy for computing this limit is to expand the 
term. To do this, you have to know a little about the so-called Binomial formula. The main fact that we will need here is that when we expand the 
term we will get
For what we are doing here, we do not need to know in detail what those trailing terms are. All that matters is that all of the other terms have at least h2 in them.
We can cancel the two xn terms above. Note that every other term in the numerator has at least one factor of h in it. This allows us to cancel the h in the denominator by removing an h from each term in the numerator.
The derivative of a constant
In this problem the function we are going to differentiate is a constant.
f(x) = c for all x
To compute the derivative, we use the definition above.
Next, we consider the function f(x) = ax and try to compute
We can simplify the last expression further by noting that
by definition. Thus,
An interesting consequence of this has to do with the definition of e. If you remember back to section 1.5, you may recall that e was defined to be the one special value of the exponential base a that has the property that the slope of the tangent line to ax at (0,1) is 1. In other words,
This leads to
or
Next we are going to focus on the second point in the strategy. We will develop a series of derivative formulas meant to implement point two.
Theorem (Sum Rule) If f(x) and g(x) are differentiable functions, then f(x) + g(x) is differentiable and
Proof Introduce the function
u(x) = f(x) + g(x)
and evaluate the limit
Substituting for u(x) we have
We can rearrange terms in the numerator
and split the limit into a sum of limits to obtain
Theorem (Constant Factors) If c is a constant and f(x) is a differentiable function, then the function g(x) = c f(x) is differentiable and
Proof By the definition of the derivative
The rules we have developed so far are sufficient to enable us to compute the derivative of any polynomial. For example, consider
By the sum rule we can say
The constant factors rule allows us to move the constant factors outside the derivatives.
The power rule and the constant derivative rule get us the rest of the way
Section 3.1: 8, 9, 17, 20, 23, 32, 42, 49, 60