in this section, we will consider the derivative function rather than just at a point. we also begin...
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In this section, we will consider the derivative function rather than just at a point. We also begin looking at some of the basic derivative rules.
Section 2.2 Derivatives of Power Functions and Polynomials
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Definition (formal)
Let f be any function. The derivative function of f is defined as:
provided the limit exists
Other notations:
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Example 1
Use the definition of derivative to find the derivative of the function .
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Example 2
Use the definition of derivative to find the derivative of the function .
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Example 3
Use the definition of derivative to find the derivative of the function .
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Example 4
Use the definition of derivative to find the derivative of the function .
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Theorem: Power Rule for Derivatives
Let n be any real number (not necessarily an integer).
Then:
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Example 5
Find the derivative of each of the following functions.
(a)
(b)
(c)
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Theorem: Constant Multiple Rule for
Derivatives
Let f be any differentiable function, let k be any constant, and let .
Then:
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Theorem: Sum Rule for Derivatives
Let f and g be any differentiable functions, and let
. Then:
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Example 6
Find the derivative of each of the following polynomials.
(a)
(b)
(c)
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Example 7
Give the equation of the tangent line to the curve at the point (1, 3).