From the source of Eve’s temptation to the cause of the Trojan War, apples have long played a key role in our legends and myths. Ripe with symbolic meanings and associations, the fruit has been used to signify temptation, wisdom, joy, fertility and youthfulness. According to popular legend, a falling apple in an English orchard in 1666 led Isaac Newton to postulate the inverse square law of gravitation.

Given the symbolic roles apples have played throughout history, it seems only fitting to focus this week’s conversation on another famous apple: Apple Inc. We’ll take a look at Apple’s revenue growth and see if we can derive the equations for business velocity and acceleration.

 iGravity

The United States officially plunged into what’s being called “The Great Recession” in December of 2007. Most of the world’s developed economies followed in 2008. By December 2010, the U.S. unemployment rate was hovering around 10% and throughout 2010, banks ended up seizing over one million U.S. homes.

Yet during the same period, Apple’s financial performance appeared to defy gravity (maybe we should call it iGravity). The company’s revenue growth from 2002 to 2011 is illustrated below.

Interestingly, in 1996, when Steve Jobs returned as CEO, the company was on the verge of bankruptcy. Today, Apple represents more than 4% of the S&P 500. Its market capitalization outweighs that of the entire materials sector, more than the telecom sector and more than the utilities sector. Generating $108 billion in revenue in one year is the equivalent of selling $3,424 worth of product every second.  Just thinking about it can make you dizzy.

As the graph’s trend line clearly illustrates, Apple is experiencing “exponential” revenue growth. One of the more interesting aspects of an exponential function is that its derivative is also an exponential function. In other words:

d[e^x]/dx = e^x

With that in mind, let’s take a look at the 1^st and 2^nd derivatives of Apple’s revenue curve. In an earlier post we defined business velocity as the time rate of change of revenue (1^st derivative) and acceleration as the time rate of change in velocity (2^nd derivative).

In this case we’ll leverage Excel’s regression capabilities to fit a 5^th order polynomial to the revenue curve.

Revenue = 0.02x⁵ – 0.4601x⁴ + 3.8843x³ – 13.411X²+ 20.972x – 5.44

Taking the first derivative of the revenue function gives us the equation for Apple’s velocity:

Velocity = 0.1x⁴ – 1.8404x³ + 11.532x² – 26.822x + 20.972

Taking the second derivative gives us the equation for Apple’s acceleration:

Acceleration = 0.4x³ – 5.5212x² + 23.064x – 26.822

The three functions are plotted together below, along with their corresponding trend lines.

One of the first things you’ll notice is that Apple’s velocity (red) and acceleration (yellow) also resemble exponential growth curves – which they should, since the derivative of an exponential function is an exponential function. Based on the model, Apple’s business velocity in 2011 was a staggering $65.552 billion per year. That’s equivalent to increasing top line revenue growth by about $2,078 per second. Its acceleration was $51.698 billion per year², which translates to $1,639 / s².

In some ways, it’s actually difficult to wrap our minds around the magnitude of these numbers. From a supply chain perspective, assuming the company can continue to meet market demand, we could reasonably predict 2012 revenues approaching $200 billion. (A number of market analysts are actually forecasting at least $157 billion.)

Even more impressive, nearly 75% of Apple’s revenues come from just two products: the iPhone (53%) and the iPad (20%). The first iPhones began shipping in June 2007 and the iPad only became available a year ago, on April 3, 2010. I don’t know about you, but I’m not aware of any other company that generates nearly 75% of its revenue from products that didn’t exist five years ago, while doing it during a global economic downturn.

Which raises the question: Given the number of consultants offering books and seminars on how to achieve “breakthrough” business performance,  why aren’t there more Apples? The question behind the question is: “What’s at the core of Apple’s success?”

A few posts ago we took at quick look at a video of Steve Jobs addressing an audience of developers and engineers, pre-iPod, iPhone and iPad.  The 1997 clip was offered as an example of someone lowering the inertial mass of an organization as the precursor to achieving a higher level of acceleration.

This week, I’d like to offer a different clip. This one features Steve Jobs clearly articulating the vision that is Apple’s “klokhuis”.  In just under seven minutes, with no props, no slides, no charts and no statistics, he reminds us all of something amazingly easy to lose sight of: despite all the technology at our disposal – the most powerful tool we have for changing the world around us is still the human voice.

At a first glance, Steve Jobs may not have fit precisely into Jim Collins’ model for Level 5 Leadership, but in looking back at Apple through the portal provided by these videos, you can clearly see the major components of Collins’ framework brought to life though his voice. Given that we now have a reasonably valid mathematical model for Apple’s acceleration, we’re in a position to begin quantifying the Force associated with that unique voice.