1. Differentiation Techniques: The Power and Sum-Difference Rules
OBJECTIVE
Differentiate using the Power Rule or the Sum-Difference Rule.
Differentiate a constant or a constant times a function.
Determine points at which a tangent line has a specified slope.
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Leibniz’s Notation:
When y is a function of x, we will also designate the
derivative, , as
which is read “the derivative of y with respect to x.”
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3. THEOREM 1: The Power Rule
1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
THEOREM 1: The Power Rule
For any real number k,
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Example 1: Differentiate each of the
following:
a) b) c) a) b) c)
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Quick Check 1
a.) Differentiate:
(i) ; (ii)
b.) Explain why , not
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Quick Check 1 solution
a.) Use the Power Rule for both parts:
(i)
(ii)
b.) The reason and not , is because is a constant, not a variable and the derivative of any constant is 0:
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Example 2: Differentiate:
a) b) a) b)
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
THEOREM 2:
The derivative of a constant function is 0. That is,
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
THEOREM 3:
The derivative of a constant times a function is the
constant times the derivative of the function. That is,
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Example 3: Find each of the following derivatives:
a) b) c) a) b)
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Example 3 (concluded): c)
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Quick Check 2
Differentiate each of the following:
a.)
b.)
c.)
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13. THEOREM 4: The Sum-Difference Rule
1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
THEOREM 4: The Sum-Difference Rule
Sum: The derivative of a sum is the sum of the
derivatives.
Difference: The derivative of a difference is the
difference of the derivatives.
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Example 4: Find each of the following derivatives:
a) b) a)
NOTE: On p. 153, it has the derivative of 7 is 10 (instead of 0).
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Example 4 (concluded): b)
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Quick Check 3
Differentiate:
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Example 5: Find the points on the graph of
at which the tangent line is horizontal.
Recall that the derivative is the slope of the tangent
line, and the slope of a horizontal line is 0. Therefore,
we wish to find all the points on the graph of f where
the derivative of f equals 0.
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Example 5 (continued):
So, for
Setting equal to 0:
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Example 5 (continued):
To find the corresponding y-values for these x-values,
substitute back into
Thus, the tangent line to the graph of
is horizontal at the points (0, 0) and (4, 32).
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Example 5 (concluded):
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Example 6: Find the points on the graph of
at which the tangent line has slope 6.
Here we will employ the same strategy as in Example
6, except that we are now concerned with where the
derivative equals 6.
Recall that we already found that
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Example 6 (continued):
Thus,
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Example 6 (continued):
Again, to find the corresponding y-values, we will
substitute these x-values into
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Example 6 (continued):
Similarly,
Thus, the tangent line to has a slope
of 6 at and
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Example 6 (concluded):
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1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Section Summary
Common forms of notation for the derivative of a function are
The Power Rule for differentiation is , for all real numbers .
The derivative of a constant is zero:
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27. 1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules
Section Summary Concluded
The derivative of a constant times a function is the constant time the derivative of the function:
The derivative of a sum (or difference) is the sum (or difference) of the derivatives of the terms:
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