Let’s recap the main ideas and
results of section 2.1, in Active Calculus on the elementary derivative rules.
So the main idea of this section and indeed this whole chapter is to use
the definition of the derivative as we developed it in chapter one.
And try to come up with ways of calculating derivative formulas quickly
by exploiting the patterns that we see when we use that limit definition.
In this section, we encounter several fundamental computational rules for
derivatives and
it's imperative that you master these basic rules as early as possible.
First of all, we saw that the derivative of any constant function
that is a function who's output is always the same value is equal to 0.
This makes sense on a number of levels since for
example of the derivative of f tells us the rate at which it is changing,
then if f is never changing we'd expect its derivative to be 0, and indeed it is.
Next, we learn that if f is a power function, that is it formally is given as
f(x) equals x to the nth power, where n is a nonzero constant power,
then the derivative of f is f prime of x equals n times x to the n -1 power.
This formula works for any nonzero power of x including negative powers and
fractional powers.
So functions like the square root of x which is x to the one-half and
one over x which is x to the negative 1 are covered by this derivative rule.
Very important thing to note here,
this rule does not apply to absolutely any function that has an exponent on it.
For example, exponential functions such as two to the x power are like
power functions except that the roles of the base and the exponent are switched.
Whereas a power function has a variable base and a constant exponent,
exponential functions have a constant base and the exponent is variable.
So for example, exponential functions do not follow this power rule,
a crucial principal in using these fast differentiation methods is to take them
very literally and do not use them on a function,
unless the function fits the rules description exactly.
So the power rule doesn't work on exponential functions,
we have a separate rule for those, namely this.
That for any positive real number a, if f(x) is the exponential function a to
the x, then f prime of x is a to the x times the natural logarithm of a.
We'll see why this derivative rule is what it is in a few more sections.
Lastly, there were two more fundamental rules in this section that work
under general conditions.
One is called the constant multiple rule, that says that for
any real number k if f(x) is differentiable,
then the derivative of k times f (x0 is equal to k times the derivative f(x).
Notice two things here,
one we are using d over dx notation that was introduced in this section.
Remember, the d over dx followed by a function just means the derivative of
that function.
And two in this rule k is a constant,
if k is a variable then this rule does not apply.
The other general rule is called the sum rule which says that if f(x) and
g(x) are differentiable of functions, then the derivative of f(x) plus g(x)
equals the derivative of f(x) plus the derivative of g(x).
The main goal for right now is to learn to use these rules both individually and
in combination with each other with absolute fluency so
that you are automatic when you are computing with them by hand.
That's the thrust of the activities in the section and more analysis and
examples of those rules at work in the remaining videos to give you a start.
