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Chapter 3 | Derivatives
3.2 | The Derivative as a Function
Learning Objectives
3.2.1 Define the derivative function of a given function. 3.2.2 Graph a derivative function from the graph of a given function. 3.2.3 State the connection between derivatives and continuity. 3.2.4 Describe three conditions for when a function does not have a derivative. 3.2.5 Explain the meaning of a higher-order derivative.
As we have seen, the derivative of a function at a given point gives us the rate of change or slope of the tangent line to the function at that point. If we differentiate a position function at a given time, we obtain the velocity at that time. It seems reasonable to conclude that knowing the derivative of the function at every point would produce valuable information about the behavior of the function. However, the process of finding the derivative at even a handful of values using the techniques of the preceding section would quickly become quite tedious. In this section we define the derivative function and learn a process for finding it. Derivative Functions The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined. We can formally define a derivative function as follows. Definition Let f be a function. The derivative function , denoted by f ′, is the function whose domain consists of those values of x such that the following limit exists: (3.9) f ′( x ) = lim h →0 f ( x + h )− f ( x ) h . A function f ( x ) is said to be differentiable at a if f ′( a ) exists. More generally, a function is said to be differentiable on S if it is differentiable at every point in an open set S , and a differentiable function is one in which f ′( x ) exists on its domain. In the next few examples we use Equation 3.9 to find the derivative of a function. Example 3.11 Finding the Derivative of a Square-Root Function
Find the derivative of f ( x ) = x .
Solution Start directly with the definition of the derivative function. Use Equation 3.1 .
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