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ABSOLUTE VALUE DERIVATIVES
Find the derivative of:
Do we treat this with the same rule as for
No.
State the rule we must apply in terms of a general function u.
What effect does the factor have?
It generates the correct algebraic sign.
What is the expression for u here? u = 3 x – 7
What is the value of
Substitute these results in the rule for the derivative above.
Will this always work out to a value of 3? No.
Which factor in can cause the expression to be negative?
Since the denominator is an absolute value, and is always positive, we need to look only at the (3 x – 7) factor in the numerator.
Find the values of x which make (3 x – 7) negative. We set (3 x – 7) < 0.
Solve for x. We get 3 x < 7, or x < 7/3 in order for the derivative of |3 x – 7| to be negative.
Check this for x = 1, which is certainly less than 7/3.
For x = 1,
So we see that whenever the value of x is less than 7/3, the derivative will be – 3.
To get more of a feel for this problem, let’s plot it in stages. First plot a related function, y = 3 x – 7 (no absolute value), on paper.
Is the slope of this graph the same everywhere? Yes, it is +3.
Is the y -coordinate positive everywhere? No.
Where is it negative? For x -values to the left of the x -intercept, the line is below the x -axis and the y -coordinate is negative.
Next we will plot y = |3 x – 7|. Will any part of the graph will be the same as the first graph? Yes.
Describe the part that is the same. Where 3 x – 7 is positive, |3 x – 7| = 3 x – 7, and the graphs will be the same.
For what values of x will this be true? For values to the right of the intercept, or
Those would be x -values along the blue part of the x -axis.
How do we need to change the part of the graph to the left of the x -intercept (for x -values along the green part of the x -axis)?
Since 3 x – 7 is negative there, we have |3 x – 7| = – (3 x – 7) for
We have
Hence the slope is – 3.
Summarize these results. Our application of the Absolute Value Rule gave us
In other words, the derivative of the absolute value is the product of a “sign factor” and the derivative of the “stuff” between the absolute value signs.
The “sign factor” is +1 or – 1.
