Tag Archives: quotient rule

Implicit Derivative Question

Yesterday, this question posted within the AP Calculus Community about implicit derivatives.

implicit

Below, I argue why the derivatives MUST be the same, show how four different variations can all be shown to give the same derivative, and provide a final conclusion.

INITIAL INTUITION

The Desmos graph of the given relation, is y^2 = \frac{x-1}{x+1} , is shown below.  Logically, it seems that even when the terms of the relation are algebraically rearranged, the graph should be invariant.   The other two forms mentioned in the Community post are on lines 2 and 3.  Lines 4, 5, and 6 show three other variations.  Here is the link to my Desmos graph allowing you to change between the forms to visually confirm the graphical invariance intuition.

implicit2.jpg

If calculus “works”, it also shouldn’t matter how one calculates a derivative.   While the forms of the derivative certainly could LOOK different,  because any point on the invariant graph has the same tangent line no matter what the form of its equation, and the derivative of a relation at a point is the slope of that invariant tangent line, then the derivative also MUST be invariant.

CALCULATING “DIFFERENT” DERIVATIVES

To show the derivatives are fundamentally all the same (as suspected by the initial post), I calculate the derivatives of the equations on lines 1 and 3 given in the initial post as well as my variations on lines 4 and 6.

LINE 1:

Using the Chain Rule on the left and the Quotient Rule on the right gives

implicit3

LINE 3:

This version is more complicated, requiring the Product Rule in addition to the earlier Chain and Quotient Rules.  In the penultimate line, I used the original equation to substitute for y^2 to transform the derivative into the same form as line 1.

implicit4.jpg

LINE 4:

This time, differentiation requires only the Chain and Product Rules.

implicit5

After the usual substitution for y^2, I multiplied both sides by (x+1) to clear the denominator and solved for y', returning the same result.

implicit6

LINE 6:

This time, the relation is solved for x, resulting in a much more complicated Quotient+Chain Rule calculation, but substituting for y^2 and changing the form leads once again to the same answer.

implicit7

Hopefully this is convincing evidence that all derivative forms can be shown to be equivalent.   If you’re still learning implicit differentiation, I encourage you to show the derivatives from the lines 2 and 5 variations are also equivalent.

CONCLUSION

So which approach is “best”?  In my opinion, it all depends on your personal comfort with algebraic manipulations.  Some prefer to just take a derivative from the given form of y^2 = \frac{x-1}{x+1}.  I avoid the more complicated quotient rule whenever I can, so the variation from line 4 would have been my approach.

The cool part is that it doesn’t matter what approach you use, so long as your algebraic manipulations are sound.  You don’t have to accept the form in which a problem is given; CHANGE IT to a form that works for you!

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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.