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 = 5t^{2} + 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, x_{1}
and x_{2}, where dx = x_{2} – x_{1 }

This means: x_{2} is at dt duration later than x_{1}.

Therefore x_{2} = 5(t + dt)^{2} + 3(t + dt)

And x_{1} = 5t^{2} + 3t

Therefore dx = x_{2} – x_{1} = 5(t + dt)^{2} +
3(t + dt) - (5t^{2} + 3t)

If we expand this we get: ~~5t~~^{2}
+ 10tdt + 5dt^{2} + ~~3t~~ + 3dt – ~~5t~~^{2} – ~~3t~~

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

Therefore dx/dt = 10t dt/dt + 5dt^{2}/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 = 5x^{2} + 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 =
ax^{2} + 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.