In high school, mathematics and physics are almost inseparable. Of the mathematics students encounter, three areas are especially closely tied to physics: trigonometric functions, vectors, and calculus.

Many fundamental definitions in physics rely heavily on the language and formulas of calculus, so I hope this topic can give you a few basic ideas to work with.

Calculus contains two major branches: differentiation and integration.

At the high-school level, you can think of differentiation and integration as actions performed on a function, such as f(x).

That is why you often see phrases such as “differentiate the function f(x) with respect to x.” In physics, however, you will more often differentiate with respect to time t—in other words, describe how quickly something changes over time, or its time rate of change.

Because differentiation and integration each deserve plenty of space, this article will first focus on the idea and mathematical meaning of differentiation.

When we differentiate a function at a particular point, we can write the definition as follows:

Definition of the derivative: f′(x) is the limit as Δx approaches zero of [f(x+Δx)−f(x)]/Δx.
Definition of the derivative: f′(x) is the limit as Δx approaches zero of [f(x+Δx)−f(x)]/Δx.

From the definition, we can see that differentiating f(x) produces another function f′(x), called the derivative of f(x).

Still, a definition alone can feel rather abstract when you are learning physics. So let us use a graph to see what differentiation is actually doing and what it means.

Graph of y=f(x): secant line A runs through Q at x and P at x+Δx; line B is the tangent at Q.
Graph of y=f(x): secant line A runs through Q at x and P at x+Δx; line B is the tangent at Q.

In the graph above, suppose P has coordinates (x+Δx, f(x+Δx)) and Q has coordinates (x, f(x)). Then

Slope of the secant through P and Q: [f(x+Δx)−f(x)]/Δx.
Slope of the secant through P and Q: [f(x+Δx)−f(x)]/Δx.

is the slope of the secant line through P and Q on the graph y=f(x)—that is, it tells us how steep the line PQ is.

This shows that the basic idea behind differentiation is finding a slope.

But the definition of the derivative adds the following notation to the slope formula:

Limit notation: as Δx approaches zero.
Limit notation: as Δx approaches zero.

What does this symbol mean, and how does it change the slope we were just discussing?

“lim” is short for “limit.” The expression Δx→0 underneath it says that the horizontal separation between P and Q approaches zero. We do not simply set Δx equal to zero in the fraction; instead, we examine what the quotient approaches as Δx becomes as small as we like.

As P moves closer and closer to Q, secant line A approaches the tangent line at Q—line B in the graph. The slope of that line is the tangent slope at Q, or at x.

So when we differentiate f at x, we are finding the slope of the tangent line to y=f(x) at that point. The value f′(x) gives that tangent slope.

Of course, drawing a graph and repeating this limiting process for every function would be far too troublesome =口=. Using the definition of the derivative, mathematicians have developed differentiation rules for different kinds of functions—for example, the familiar power rule often remembered as “bring down the exponent, then subtract one.”

High-school physics constantly asks how quickly physical quantities change with time, so differentiation appears very often. Later high-school mathematics is divided into different content according to the curriculum; this article focuses only on the basic ideas of differentiation most often used in high-school physics. It really does not have to be that frightening, so do not worry :D.