The Power Rule

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The Power Rule

• If n is any real number, then

In other words, to find the derivative, take the
exponent of x, move it to the front of the
function, then decrease the exponent by one.
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Proof of the power rule
Expanding this using the binomial theorem
Notice that the first and last terms cancel,
leaving
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Practice using the power rule
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Practice using the power rule
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Now try finding f(x) given
We can evaluate this as 2x2 New rule, the
constant multiple rule
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What is the derivative of a constant?
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Problem Horizontal Tangents

• Given the following equation, find all the points
  on which the tangent would be horizontal

The tangent is horizontal whenever the slope is 0
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Problem Acceleration at a given time

• The position of a particle is given as the
  following function. At 2 seconds, what is the
  acceleration of that particle?

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Exponential functions

• Finding the derivative of an exponential function
  is trickier than in power functions

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Notice that ax is not dependent on h, so we can
move it to the front
Using the graphing calculator table of values,
notice that the limit is the derivative of f at
0, thus
Another way of looking at this the rate of
change of any exponential function is
proportional to the function itself.
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Another way of looking at this the rate of
change of any exponential function is
proportional to the function itself.
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Finding the derivative of
If we substitute values close to 0, we get the
fact that
So ..
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If we choose other values other than 2, we must
calculate by using

• If a 2, then k .693
• a 3, then k 1.099
• a 4, then k 1.396
• a 5, then k 1.609
• etc., etc., etc.,

So when is k 1?
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So when is k 1?

• Since we would like to know when

Solve for h
And take small values of h, we find that a
..
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2.718281828459045

• e

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The formula for the derivative of ax

• We saw a little while ago that the derivative of
  2x is k2x, and we then used limits to find the
  constant.
• In this case, k was .693, which is also ln(2)
• Using these limits and a long proof shown on p.
  119, we can simplify this all to say

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So what does this mean?

• If a 2, then k .693
• a 3, then k 1.099
• a 4, then k 1.396
• a 5, then k 1.609
• and
• a e, then k 1
• Which means that the derivative of ex is ex
• As the derivative of this exponential function is
  so easy, it is used frequently in problems as the
  base.

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Back to e

• Apply this to the value e

We know that ln e has a value of 1, so it works!
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Definition of the Number e

• e is the number such that

In other words, of all the possible exponential
functions, this is the only one whose tangent
line at (0,1) has a slope of exactly 1.