There is one last topic to discuss in this section. Taking the derivatives of some complicated
functions can be simplified by using logarithms. This is called logarithmic differentiation.
It’s easiest to see how this works in an example.
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Example 1 Differentiate
the function.
Solution
Differentiating this function could be done with a product
rule and a quotient rule. However,
that would be a fairly messy process.
We can simplify things somewhat by taking logarithms of both sides.
Of course, this isn’t really simpler. What we need to do is use the properties of
logarithms to expand the right side as follows.
This doesn’t look all that simple. However, the differentiation process will
be simpler. What we need to do at this
point is differentiate both sides with respect to x. Note that this is
really implicit differentiation.
To finish the problem all that we need to do is multiply
both sides by y and the plug in for
y since we do know what that is.
Depending upon the person, doing this would probably be
slightly easier than doing both the product and quotient rule. The answer is almost definitely simpler
than what we would have gotten using the product and quotient rule.
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So, as the first example has shown we can use logarithmic
differentiation to avoid using the product rule and/or quotient rule.
We can also use logarithmic differentiation to
differentiation functions in the form.
Let’s take a quick look at a simple example of this.
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Example 2 Differentiate
Solution
We’ve seen two functions similar to this at this point.
Neither of these two will work here since both require
either the base or the exponent to be a constant. In this case both the base and the exponent
are variables and so we have no way to differentiate this function using only
known rules from previous sections.
With logarithmic differentiation we can do this
however. First take the logarithm of
both sides as we did in the first example and use the logarithm properties to
simplify things a little.
Differentiate both sides using implicit differentiation.
As with the first example multiply by y and substitute back in for y.
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Let’s take a look at a more complicated example of this.
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Example 3 Differentiate
Solution
Now, this looks much more complicated than the
previous example, but is in fact only slightly more complicated. The process is pretty much identical so we
first take the log of both sides and then simplify the right side.
Next, do some
implicit differentiation.
Finally, solve
for and substitute back in for y.
A messy answer but there it is.
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We’ll close this section out with a quick recap of all the
various ways we’ve seen of differentiating functions with exponents. It is important to not get all of these
confused.
It is sometimes easy to get these various functions confused
and use the wrong rule for differentiation.
Always remember that each rule has very specific rules for where the
variable and constants must be. For
example, the Power Rule requires that the base be a variable and the exponent
be a constant, while the exponential function requires exactly the opposite.
If you can keep straight all the rules you can’t go wrong
with these.