To sketch: The graph of and then below it.
The formula for from its graph is, .
The given function is .
From the graph of , it is clear that the slope is always positive. Thus, the derivative of the graph is always positive.
Draw the tangent to the graph at and the slope of the graph is computed as follows,
Recall the fact that, the slope of the function at the point is equal to the derivative of the function at that point.
Therefore, at .
Thus, the graph of the function passes through the point .
From the point A to left, the slope is increasing slowly. Thus the graph of the function in this section will rise slowly.
From the point A to right, the slope increases rapidly. Thus the graph of the function in this section will rise highly.
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 a natural exponential function.
Thus, the derivative formula is, .
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