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In this section we’re going to introduce a notation that
we’ll be seeing quite a bit in the next chapter. We will also look at an application of this
new notation.
Given a function 
we call dy
and dx differentials and the
relationship between them is given by,
Note that if we are just given then the differentials are df and dx and we compute them the same manner.
Let’s compute a couple of differentials.
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Example 1 Compute
the differential for each of the following.
(a)
(b)
(c)
Solution
Before working any of these we should first discuss just
what we’re being asked to find here.
We defined two differentials earlier and here we’re being asked to
compute a differential.
So, which differential are we being asked to compute? In this kind of problem we’re being asked
to compute the differential of the function.
In other words, dy for the
first problem, dw for the second
problem and df for the third
problem.
Here are the solutions.
Not much to do here other than take a derivative and don’t forget to
add on the second differential to the derivative.
(a)
(b)
(c)
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There is a nice application to differentials. If we think of as the change in x then is the change in y corresponding to the change in x. Now, if is small we can assume that . Let’s see an illustration of this idea.
We can use the fact that in the following way.
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Example 3 A
sphere was measured and its radius was found to be 45 inches with a possible
error of no more that 0.01 inches.
What is the maximum possible error in the volume if we use this value
of the radius?
Solution
First, recall the equation for the volume of a sphere.
Now, if we start with and use then should give us maximum error.
So, first get the formula for the differential.
Now compute dV.
The maximum error in the volume is then approximately
254.47 in3.
Be careful to not assume this is a large error. On the surface it looks large, however if
we compute the actual volume for we get . So, in comparison the error in the volume
is,
That’s not much
possible error at all!
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