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# Derivatives

# Graphs of *f *( *x* ) and *f* ' ( *x* )

From a graph of a function *f*(*x*) we can make a sketched graph of its derivative *f ' *(*x*). To do this, we use some things we talked about earlier.

- If
*f*is decreasing, its slope (and hence its derivative) is negative. If*f*is increasing, its slope (and hence its derivative) is positive.

- From drawing tangent lines to
*f*, we can compare relative values of the derivative and tell where the derivative is greatest.

- If
*f*is a line, its slope is constant.

**Be Careful:** The word "it" is dangerous.

Look at these two sentences:

- It's increasing, so it's positive.

*f*is increasing, so*f '*is positive.

The first sentence is unclear. What does "it" mean? There's no way to know. Whenever we use the words "increasing, decreasing, positive, negative," that we are clear about what (*f*, *f '*, or something else?) is increasing, decreasing, positive, or negative.