Differentiation Formulas
In the first section of this chapter we saw the
definition of the derivative
and we
computed a couple of derivatives using the definition. As we saw in those examples
there was a fair amount of work involved in computing the limits and the functions
that we worked with were not terribly complicated.
For more complex functions using the definition of the derivative would be an almost
impossible task. Luckily for us we won’t have to use the definition terribly often. We
will have to use it on occasion, however we have a large collection of formulas and
properties that we can use to simplify our life considerably and will allow us to avoid
using the definition whenever possible.
We will introduce most of these formulas over the course of the next several sections.
We will start in this section with some of the basic properties and formulas. We will
give the properties and formulas in this section in both “prime” notation and
“fraction” notation.
Properties
1)
OR
In other words, to differentiate a sum or difference all we need to do is differentiate the
individual terms and then put them back together with the appropriate signs. Note as well
that this property is not limited to two functions.
See the
Proof of Various Derivative Formulas
section of the Extras chapter to see the
proof of this property. It’s a very simple proof using the definition of the derivative.
2)
OR
,
c
is any number
In other words, we can “factor” a multiplicative constant out of a derivative if we need to.
See the
Proof of Various Derivative Formulas
section of the Extras chapter to see the
proof of this property.
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Note that we have not included formulas for the derivative of products or quotients of
two functions here. The derivative of a product or quotient of two functions is not the
product or quotient of the derivatives of the individual pieces. We will take a look at
these in the next section.
Next, let’s take a quick look at a couple of basic “computation” formulas that will
allow us to actually compute some derivatives.
Formulas
1)
If
then
O
R
The derivative of a constant is zero. See the
Proof of Various Derivative
Formulas
section of the Extras chapter to see the proof of this formula.
2)
If
then
OR
,
n
is any number.
This formula is sometimes called the
power rule
. All we are doing here is bringing the
original exponent down in front and multiplying and then subtracting one from the original
exponent.
Note as well that in order to use this formula
n
must be a number, it can’t be a variable.
Also note that the base, the
x
, must be a variable, it can’t be a number. It will be tempting
in some later sections to misuse the Power Rule when we run in some functions where the
exponent isn’t a number and/or the base isn’t a variable.
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 Fall '08
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 Derivative, Formulas, Various Derivative Formulas, Derivative Formulas section

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