# Tag Archives: chain rule

## Calculus Derivative Rules

Over the past few days I’ve been rethinking my sequencing of introducing derivative rules for the next time I teach calculus.  The impetus for this was an approach I encountered in a Coursera MOOC in Calculus I’m taking this summer to see how a professor would run a Taylor Series-centered calculus class.

Historically, I’ve introduced my high school calculus classes to the product and quotient rules before turing to the chain rule.  I’m now convinced the chain rule should be first because of how beautifully it sets up the other two.

Why the chain rule should be first

Assuming you know the chain rule, check out these derivations of the product and quotient rules.  For each of these, $g_1$ and $g_2$ can be any differentiable functions of x.

PRODUCT RULE:  Let $P(x)=g_1(x) \cdot g_2(x)$.  Applying a logarithm gives,

$ln(P)=ln \left( g_1 \cdot g_2 \right) = ln(g_1)+ln(g_2)$.

Now differentiate and rearrange.

$\displaystyle \frac{P'}{P} = \frac{g_1'}{g_1}+\frac{g_2'}{g_2}$
$\displaystyle P' = P \cdot \left( \frac{g_1'}{g_1}+\frac{g_2'}{g_2} \right)$
$\displaystyle P' = g_1 \cdot g_2 \cdot \left( \frac{g_1'}{g_1}+\frac{g_2'}{g_2} \right)$
$P' = g_1' \cdot g_2+g_1 \cdot g_2'$

QUOTIENT RULE:  Let $Q(x)=\displaystyle \frac{g_1(x)}{g_2(x)}$.  As before, apply a logarithm, differentiate, and rearrange.

$\displaystyle ln(Q)=ln \left( \frac{g_1}{g_2} \right) = ln(g_1)-ln(g_2)$
$\displaystyle \frac{Q'}{Q} = \frac{g_1'}{g_1}-\frac{g_2'}{g_2}$
$\displaystyle Q' = Q \cdot \left( \frac{g_1'}{g_1}-\frac{g_2'}{g_2} \right)$
$\displaystyle Q' = \frac{g_1}{g_2} \cdot \left( \frac{g_1'}{g_1}-\frac{g_2'}{g_2} \right)$
$\displaystyle Q' = \frac{g_1'}{g_2}-\frac{g_1 \cdot g_2'}{\left( g_2 \right)^2} = \frac{g_1'g_2-g_1g_2'}{\left( g_2 \right)^2}$

The exact same procedure creates both rules. (I should have seen this long ago.)

Proposed sequencing

I’ve always emphasized the Chain Rule as the critical algebra manipulation rule for calculus students, but this approach makes it the only rule required.  That completely fits into my overall teaching philosophy:  learn a limited set of central ideas and use them as often as possible.  With this, I’ll still introduce power, exponential, sine, and cosine derivative rules first, but then I’ll follow with the chain rule.  After that, I think everything else required for high school calculus will be a variation on what is already known.  That’s a lovely bit of simplification.

I need to rethink my course sequencing, but I think it’ll be worth it.