Transcript 2.2

Chapter Two: Section Two
Basic Differentiation Rules and Rates of Change
Chapter Two: Section Two
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Remember from our last section that the complex
definition of the derivative will give us:
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Either a simple numerical value if we decide in advance a
target domain value at which to construct our tangent line
or
An algebraic formula that will then find the slope of the
tangent line at any domain value for the function.
Chapter Two: Section Two
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What we need to remind ourselves of constantly,
and it is one of the reasons why I am emphasizing
the definition of the derivative, is we are finding a
slope whenever we talk about the derivative of a
function.
This idea should inform some of the following
differentiation rules that are presented clearly in
section two of this chapter.
Chapter Two: Section Two
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The constant rule – The derivative of a constant is
zero.
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Why would this be true?
What do we know about functions that are constants, such
as the function y = 5?
Why woould the derivative of such a function be zero?
Chapter Two: Section Two
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The power rule – If n is a rational number, then the
function f(x) = xn is a differentiable function and the
derivative of the function is nxn-1.
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Can you convince yourself that this rule makes sense?
What should the derivative of a linear function look like?
What degree would that function be?
Think about the derivative of a parabola. What should it
look like according to the power rule? Can you convince
yourself that this is true?
Chapter Two: Section Two
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The constant multiple rule - If f is a differentiable
function and c is a constant, then the function cf is
a differentiable function whose derivative is simply c
times the derivative of f.
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Would a similar rule apply to the function f + c ?
Why should this rule work? Think again about the simple
parabola whose equation is y = x2. Why would its derivative
be different than the derivative of the function y = 3x2 ?
Chapter Two: Section Two
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The sum and difference rules – The sum or
difference of two differentiable functions f and g are
themselves differentiable functions. The derivative of
the function f + g is simply the sum of the
derivatives of f and g. Similarly, the derivative of the
difference function f – g is the difference of the
derivatives of the individual functions f and g.
Chapter Two: Section Two
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Finally, you are presented with the rather amazing
fact that the cosine and sine curves are related
through differentiation as well. We will look at the
verification of these facts using our trigonometric
identities for angle addition and we will look at this
using an EXCEL spreadsheet as well.