College Math Teaching

September 8, 2018

Proving a differentiation formula for f(x) = x ^(p/q) with algebra

Filed under: calculus, derivatives, elementary mathematics, pedagogy — collegemathteaching @ 1:55 am

Yes, I know that the proper way to do this is to prove the derivative formula for f(x) = x^n and then use, say, the implicit function theorem or perhaps the chain rule.

But an early question asked students to use the difference quotient method to find the derivative function (ok, the “gradient”) for f(x) = x^{\frac{3}{2}} And yes, one way to do this is to simplify the difference quotient \frac{t^{\frac{3}{2}} -x^{\frac{3}{2}} }{t-x} by factoring t^{\frac{1}{2}} -x^{\frac{1}{2}} from both the numerator and the denominator of the difference quotient. But this is rather ad-hoc, I think.

So what would one do with, say, f(x) = x^{\frac{p}{q}} where p, q are positive integers?

One way: look at the difference quotient: \frac{t^{\frac{p}{q}}-x^{\frac{p}{q}}}{t-x} and do the following (before attempting a limit, of course): let u= t^{\frac{1}{q}}, v =x^{\frac{1}{q}} at which our difference quotient becomes: \frac{u^p-v^p}{u^q -v^q}

Now it is clear that u-v is a common factor..but HOW it factors is essential.

So let’s look at a little bit of elementary algebra: one can show:

x^{n+1} - y^{n+1} = (x-y) (x^n + x^{n-1}y + x^{n-2}y^2 + ...+ xy^{n-1} + y^n)

= (x-y)\sum^{n}_{i=0} x^{n-i}y^i (hint: very much like the geometric sum proof).

Using this:

\frac{u^p-v^p}{u^q -v^q} = \frac{(u-v)\sum^{p-1}_{i=0} u^{p-1-i}v^i}{(u-v)\sum^{q-1}_{i=0} u^{q-1-i}v^i}=\frac{\sum^{p-1}_{i=0} u^{p-1-i}v^i}{\sum^{q-1}_{i=0} u^{q-1-i}v^i} Now as

t \rightarrow x we have u \rightarrow v (for the purposes of substitution) so we end up with:

\frac{\sum^{p-1}_{i=0} v^{p-1-i}v^i}{\sum^{q-1}_{i=0} v^{q-1-i}v^i}  = \frac{pv^{p-1}}{qv^{q-1}} = \frac{p}{q}v^{p-q} (the number of terms is easy to count).

Now back substitute to obtain \frac{p}{q} x^{\frac{(p-q)}{q}} = \frac{p}{q} x^{\frac{p}{q}-1} which, of course, is the familiar formula.

Note that this algebraic identity could have been used for the old f(x) = x^n case to begin with.

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