2.2 The Derivative as a Function
In section 2.1 we focused on the derivative at a point (f '(a)). In this section we look at the derivative of a function in general, i.e. f '(x). Replacing a in the definition of f '(a) with x gives us the derivative of f,
which is a function, rather than a number. We interpret this function geometrically as the slope of the tangent line to the graph of f at the point (x, f(x)). Example: The graphs of two functions are shown below. Use the given graphs to sketch the graphs of their derivatives.
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Use the definition of the derivative to find the derivative of the given function. Then compare the graphs of f(x) and f '(x).
f(x) = x2 ­ 4x
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Use the definition of the derivative to find the derivative of the given function. Then state the domains of both functions.
3
Find f ' if and compare the graphs of f and f '.
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There are several different ways of denoting the derivative of f(x) that you may see.
The symbols D and d/dx are called differentiation operators because they indicate the operation of differentiation, which is the process of calculating a derivative. The symbol dy/dx, which was introduced by Leibniz, is a useful notation and can also be used to indicate the value of the derivative at a, (f '(a)), as follows:
or
So, for example, if you are given the function y = 5x + 7, you might be asked to find dy/dx or y', both of which would mean the derivative of the function with respect to x.
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Definition: A function f is differentiable at a if f '(a) exists. It is differentiable on an open interval (a, b) if it is differentiable at every number in the interval. If f is differentiable at a, then f is continuous at a. However, the converse is not true. Just because f is continuous at a does not make it differentiable at a. (A prime example is f (x) = |x|. It is continuous everywhere but not differentiable at 0.
There are three cases in which a function is not differentiable:
Use the graphs of the functions below to sketch the graphs of their derivatives.
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Higher Derivatives:
The derivative of f '(x) is denoted f ''(x) (called the second derivative) and is defined as
The derivative of f ''(x) is denoted f '''(x) (called the third derivative) and is defined as
etc.
Alternate notations for the second and third derivatives, respectively, are as follows:
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Given f(x) = x2 ­ 4x, find f ', f '', and f '''. (See p. 2 for f '.)
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