To sketch: The graph of and then below it and guess the formula for .
The formula for from its graph is .
The given function is .
Recall the fact that, the derivative of a function at is zero if the function has horizontal tangent at the point .
From the graph, it is observed that the graph has a horizontal tangent at three points. Let the points be A, B, C and D. At these points, the derivative will be zero.
The AB section of the graph has negative slope which implies that the derivative will be negative in the section.
The BC section of the graph has positive slope which implies that the derivative will be positive in the section.
The CD section of the graph has negative slope which implies that the derivative will be negative in the section.
Use the online graphing calculator to draw the graph of and use the above information to trace the graph of as shown below in Figure 1.
From the Figure 1, it is observed that the graph of the derivative function seems to be the cosine function.
Thus, the formula of the derivative function is, .
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