, below, find the third derivative,
:
11. Too many numbers?
Sometimes a student (or anyone solving a math problem) will get bogged down with
all the numbers and lose track of the big picture of the problem. For some
students, converting the numerical problem into a symbolic one will lead to a
simpler and more elegant solution that the student (and tutor) can readily see.
Caution – this technique works well for students who are comfortable with the idea of substituting symbols for numbers at the outset of a problem, and then re-substituting back the numbers for the symbols. Most students see that the substitution technique does not change the problem in a fundamental way, but this is not so for all students. For those students who would interpret the problem (3 + 4) as fundamentally different from the problem (A + B) where A = 3 and B = 4, they should NOT be tutored using this technique.
For those students for whom it will be suitable, here is an example. Note that this technique is most helpful when the original problem contains nasty decimals or fractions.
Given the function
, below, find the third derivative,
:
Method 1: without using symbolic substitution:
The first derivative is:
which reduces to:
and then:
From this, the second derivative is written as:
And finally, the third derivative is written as:
What a mess!
Method 2: with using symbolic substitution:
Let a = -0.245
Let b = 1.334455
Let c = -0.0262
The problem is now:
The first derivative is much more easily written directly as:
which simplifies to:
and then:
From this, the second derivative is easily written as:
And finally, the third derivative is written as:
As an extra added bonus, with a little introspection, we can see that the
nth derivative would be this:
Those messy numbers can then be substituted back in for the symbols a, b, and c to get the final answer to the problem. I fact, most instructors that I know would be perfectly happy to have the symbolic solution along with the list of substitutions.
So you be the judge. Which calculation is easier to perform and less prone to arithmetic errors? Your two choices are to use symbols a, b, and c, or trying to carry all those decimal quantities.