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I know $f'(1/2)= 1$ because the professor did that part of the problem in class. But unfortunately I just cannot see the logic to complete the rest of the problem. This problem is supposed to be really easy.

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I had to rollback to include the image link, because the question/answer made no sense without it. – Emily Nov 2 '12 at 18:04

Try applying the chain rule. For example, $\dfrac{d}{dx}{f\left(e^{x}\right)}=e^x\cdot f'(t)\lvert_{t=e^x}\;.$

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The derivative of a function at a point is essentially the slope of the line that best approximates the function at that point. The idea to find the derivative of a function by its graph would be to pick a line that approximates the graph at a point very well. The derivative is then the slope of this line. To find a line that approximates a function well at some point, you can pick two points near the point you want to approximate the function at and draw a line through them. The derivative is the slope of the line that you get when you make the two points as close as possible to the point you approximating the function at. Try doing this with the graph you have and see what happens.

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