The genius of differential calculus

Newton and Leibniz are both credited as independent ‘inventors’ of calculus but I would argue that it was at least as much discovery as invention, because, at its heart, differential calculus delivers the seemingly impossible.

Calculus was arguably the greatest impetus to physics in the scientific world. Newton’s employment of calculus to give mathematical definition and precision to motion was arguably as significant to the future of physics as his formulation of the General Theory of Gravity. Without calculus, we wouldn’t have Einstein’s Theory of Relativity and we wouldn’t have Schrodinger’s equation that lies at the heart of quantum mechanics. Engineers, the world over, routinely use calculus in the form of differential equations to design most of the technological tools and infrastructure we take for granted.

Differential calculus is best understood in its application to motion in physics and to tangents in Cartesian analytic geometry. In both cases, we have mathematics describing a vanishing entity, and this is what gives calculus its power, and also makes it difficult for people to grasp, conceptually.

Calculus can freeze motion, so that at any particular point in time, knowing an object’s acceleration (like a free-falling object under gravity, for example) we can determine its instantaneous velocity, and knowing its velocity we can determine its instantaneous position. It’s the word ‘instantaneous’ that gives the game away.

In reality, there is no ‘instantaneous’ moment of time. If you increase the shutter speed of a camera, you can ‘freeze’ virtually any motion, from a cricket ball in mid-flight (baseball for you American readers) to a bullet travelling faster than the speed of sound. But the point is that, no matter how fast the shutter speed, there is still a ‘duration’ that the shutter remains open. It’s only when one looks at the photographic record, that one is led to believe that the object has been captured at an instantaneous point in time.

Calculus does something very similar in that it takes a shorter and shorter sliver of time to give an instantaneous velocity or position.

I will take the example out of Keith Devlin’s excellent book, The Language of Mathematics; Making the invisible visible, of a car accelerating along a road:

x = 5t² + 3t

The above numbers are made up, but the formulation is correct for a vehicle under constant acceleration. If we want to know the velocity at a specific point in time we differentiate it with respect to time (t).

The differentiated equation becomes dx/dt, which means that we differentiate the distance (x) with respect to time (wrt t).

To get an ‘instantaneous’ velocity, we take smaller and smaller distances over smaller and smaller durations. So dx/dt is an incrementally small distance divided by an incrementally small time, so mathematically we are doing exactly the same as what the camera does.

But dx occurs between 2 positions, x₁ and x₂, where dx = x₂ – x₁

This means:  x₂ is at dt duration later than x₁.

Therefore  x₂ = 5(t + dt)² + 3(t + dt)

And x₁ = 5t² + 3t

Therefore  dx = x₂ – x₁ = 5(t + dt)² + 3(t + dt) - (5t² + 3t)

If we expand this we get:  5t² + 10tdt + 5dt² + 3t + 3dt – 5t² – 3t

{Remember: (t + dt)² = t² + 2tdt + dt²}

Therefore dx/dt = 10t dt/dt + 5dt²/dt + 3dt/dt

Therefore dx/dt = 10t + 3 + 5dt

The sleight-of-hand that allows calculus to work is that the dt term on the RHS disappears so that dx/dt gives the instantaneous velocity at any specified time t. In other words, by making the duration virtually zero, we achieve the same result as the recorded photo, even though zero duration is physically impossible.

This example can be generalised for any polynomial: to differentiate an equation of the form, 
y = ax^b

dy/dx = bax^(b-1)  which is exactly what I did above:

If y = 5x² + 3x

Then dy/dx = 10x + 3

The most common example given in text books (and even Devlin’s book) is the tangent of a curve, partly because one can demonstrate it graphically.

If I was to use an equation of the form y = ax² + bx + c, and differentiate it, the outcome would be exactly the same as above, mathematically. But, in this case, one takes a smaller and smaller x, which corresponds to a smaller and smaller y or f(x). (Note that f(x) = y, or f(x) and y are synonymous in this context). The slope of the tangent is dy/dx for smaller and smaller increments of dx. But at the point where the tangent’s slope is calculated, dx becomes infinitesimal. In other words, dx ultimately disappears, just like dt disappeared in the above worked example.

Devlin also demonstrates how integration (integral calculus), which in Cartesian analytic geometry calculates the area under a curve f(x), is the inverse function of differential calculus. In other words, for a polynomial, one just does the reverse procedure. If one differentiates an equation and then integrates it one simply gets the original equation back, and, obviously, vice versa.