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For the function below, which is greater: f'(0) or f'(1)?

Draw the tangent lines to f at 0 and 1. Which is greater?

The tangent line to f at 0 is steeper than the tangent line to f at 1, and so has a greater slope.
This means f'(0) > f'(1).

Example 2

For the function shown below, which is greater: f'(-1) or f'(1)?

We draw the tangent lines to f at the two points in question:

Since both slopes are negative, the one that is "less negative" will be closer to zero, and so will be greater. The tangent line to f at 1 has the "less negative" slope, so f'(1) > f'(-1).