Differentiation Techniques

Maria Gaetana Agnesi

So, no one wants to do complicated limits to find derivatives. There are easier ways of course. There are a number of quick ways (rules, formulas) for finding derivatives of the Elementary Functions and their compositions. Here are some ways to introduce these rules; these are the subject of this week’s review of past posts.

The Derivative I        Guessing the derivatives from the definition

The Derivative II      Using difference Quotient to graph and guess

The Derivative Rules I    The Power Rule

The Derivative Rules II       Another approach to the Product Rule from my friend Paul Foerster

The Derivative Rules III     The Quotient Rule developed using the Power Rule, an approach first suggested  by Maria Gaetana Agnesi (1718 – 1799) who was helping her brother learn the calculus.

Next week: The Chain Rule.

Power Rule Implies Chain Rule

Having developed the Product Rule $d\left( uv \right)=u{v}'+{u}'v$ and the Power Rule $\frac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}$ for derivatives in your class, you can explore similar rules for the product of more than two functions and suddenly the Chain Rule will appear.

For three functions use the associative property of multiplication with the rule above:

$d\left( uvw \right)=d\left( \left( uv \right)w \right)=u\cdot v\cdot dw+w\cdot d(uv)=u\cdot v\cdot dw+w\left( udv+vdu \right)$

So expanding with a slight change in notation:

$d\left( uvw \right)=uv{w}'+u{v}'w+u'vw$

For four factors there is a similar result:

$d\left( uvwz \right)=uvw{z}'+uv{w}'z+u{v}'wz+{u}'vwz$

Exercise: Let ${{f}_{i}}$ for $i=1,2,3,...,n$ be functions. Write a general formula for the derivative of the product ${{f}_{1}}{{f}_{2}}{{f}_{3}}\cdots {{f}_{n}}$ as above and in sigma notation

$d\left( {{f}_{1}}{{f}_{2}}{{f}_{3}}\cdots {{f}_{n}} \right)={{f}_{1}}{{f}_{2}}{{f}_{3}}\cdots {{{f}'}_{n}}+{{f}_{1}}{{f}_{2}}{{{f}'}_{3}}\cdots {{f}_{n}}+{{f}_{1}}{{{f}'}_{2}}{{f}_{3}}\cdots {{f}_{n}}+\cdots +{{{f}'}_{1}}{{f}_{2}}{{f}_{3}}\cdots {{f}_{n}}$

$\displaystyle d\left( {{f}_{1}}{{f}_{2}}{{f}_{3}}\cdots {{f}_{n}} \right)=\sum\limits_{i=1}^{n}{\frac{{{f}_{1}}{{f}_{2}}{{f}_{3}}\cdots {{f}_{n}}}{{{f}_{i}}}{{{{f}'}}_{i}}}$

This idea may now be used  to see the Chain Rule appear. Students may guess that $d{{\left( f \right)}^{4}}=4{{\left( f \right)}^{3}}$, but wait there is more to it.

Write ${{\left( f \right)}^{4}}=f\cdot f\cdot f\cdot f\text{ }$. Then from above

$d{{\left( f \right)}^{4}}=d\left( f\cdot f\cdot f\cdot f\text{ } \right)=f\cdot f\cdot f\cdot {f}'+f\cdot f\cdot {f}'\cdot f+f\cdot {f}'\cdot f\cdot f+{f}'\cdot f\cdot f\cdot f$

$d{{\left( f \right)}^{4}}=4{{\left( f \right)}^{3}}{f}'\text{ }$

Looks just like the power rule, but there’s that “extra” ${f}'$. Now you are ready to explain about the Chain Rule in the next class.

The Derivative Rules I

The time is approaching when you will want and need to find derivatives quickly. I am afraid that, with the exception of the product rule, I have no particularly clever ideas of how to how to teach this.

I am inclined to offer some explanation, short of a lot of proofs, to students as to why the rules and procedure are what they are. To that end I would start with some simple formulas using the (limit) definition of derivative.

• The derivative of constant times a function is the constant times the derivative of the function. This is easy enough to show from the definition since constants may be factors out of limits
• Likewise, the derivative of a sum or difference of functions is the sum of difference of the derivatives of the functions. This too follows easily from the properties of limits.
• For powers, keeping in mind the guesses from mention previously in the previous two posts “The Derivative I and II”, I suggest the method that all the books show. For example to find the derivative of x3 write

$\displaystyle \underset{h\to 0}{\mathop{\lim }}\,\frac{{{\left( x+h \right)}^{3}}-{{x}^{3}}}{h}=\underset{h\to 0}{\mathop{\lim }}\,\frac{{{x}^{3}}+3{{x}^{2}}h+3x{{h}^{2}}+{{h}^{3}}-{{x}^{3}}}{h}$

$\displaystyle =\underset{h\to 0}{\mathop{\lim }}\,\left( 3{{x}^{2}}h+3x{{h}^{2}} \right)=3{{x}^{2}}$

And perhaps a one or two more until the students are convinced of the pattern.

• For trigonometric functions follow your textbook: use the definition and the formula for the sine of the sum of two numbers along with the two special limits.

The next post will concern the product and quotient rules.