Calculus can have a frightening ring to some people, and though it is considered a “higher” form of mathematics, it may not be as difficult to grasp as you think it is. Calculus is the branch of mathematics that is concerned with situations and equations that change over time, as opposed to, well, ones that don’t. Common terms you’ll hear within calculus include function, derivative, integral, differentiation, integration, limit, etc.

A good understanding of calculus is important for most sciences and leads to more advanced concepts of mathematics. In this article it will be assumed the reader is somewhat familiar with certain pre-calculus principles such as coordinate systems, functions, and algebra. There are two main fields of (single-variable) calculus: differential calculus and integral calculus.

**Differential Calculus – the Difference Quotient, and the Power Rule for Polynomials**

This is the branch of calculus that is used to calculate the derivative, or slope, of a function. This process is called differentiation. The derivative at a given point on the curve is simply a measure of the rate of change at that point. Slope is defined as the rise over run, and in the same way, the derivative is the rise over run of a line that is tangent to the point being measured.

- A common example used to explain differentiation is the speedometer. As a vehicle accelerates, the speedometer will display what is called the instantaneous velocity of the vehicle, that is, the speed of the vehicle at any given time. Measuring the changes in a given function is what differentiation is all about.

Another way to look at it is with a line that initially bridges two points on the curve, called the secant line (see images above). The secant line simply averages the rate of change between the two given points, but of course, isn’t all that accurate. But as the two points become arbitrarily closer to each other, the secant line will become increasingly more accurate until the two points eventually merge, at which point the secant line becomes the tangent line.

- Click the image to watch the tangent line follow the curve.

The derivative of all points in the function’s domain produces a new function often denoted as *f’*, or f-prime. As an example, if we use the function *f(x) = x ^{2}*, its derivative would be

*f'(x) = 2x*. This means that the derivative at any point on the curve is equal to double the value of

*x*at that point. You’re probably thinking “that’s great, but how do you perform that differentiation?”.

Differentiation is typically computed using one of various “rules” such as the difference quotient or the power rule for polynomials. We’ll start with the difference quotient:

However, before we can solve this we need to know about **limits**. Using the secant line through two given points, *x* and *x+h*, we get an average slope. If *x* is the point of interest, then our average slope will become increasingly close to the “true” derivative as *h* tends to zero. But being that division by zero is impossible, you can’t just replace *h* with zero.

This is where the concept of limits comes into play. A limit takes a value, for example h, and considers its behavior through each value of *h* and arrives at a value for when it reaches zero – just as the secant line in the images above can be thought of to gradually move closer to the tangent line at *x*. Now let’s move on to the differentiation. You can take any value of *x* and plug it into the following equation and you should get a value corresponding to *2x*, but as an example we’ll use *f'(4)*:

Proper understanding of the difference quotient requires knowledge of binomial-squaring. The squaring of a binomial always results in three terms: the square of the first term, twice the product of the two terms, and the square of the second term. With this in mind you can see clearly how in this case, the difference quotient can be simplified to equal 8, which is *2x* as expected.

**The Power Rule for Polynomials**

This rule is comparatively easier to understand, and states that the derivative of the function *f(x)=x ^{n}* is

*f'(x)=nx*, or,

^{n-1}*(x*. As you can see, by replacing

^{n})’=nx^{n-1}*n*with

*2*, you get the

*x*function, and as expected we get the same

^{2}*2x*value as the derivative

*(f’)*. To clarify:

read, “*the derivative of x to the power of n is equal to n-x to the power of n minus 1*“. For a more comprehensive guide on how to find derivatives, read my article All About Derivatives – How to Calculate them Easily.

**Integral Calculus – the Power Rule for Integration and the Fundamental Theorem of Calculus**

- Along the lines of the speedometer example used to explain differentiation above, integration can be thought of as an
**odometer**. As the curve of a function traces out the position of a vehicle with respect to time, a line tangent to the curve will mark its instantaneous velocity (derivative), and the area underneath the curve between two arbitrary points on the time axis represent the total distance traveled (integral).

Within integral calculus, there are two types: the definite integral and indefinite integral. Integral calculus is used to determine the area under a given function, within the prescribed interval. The process of computing integrals is called integration.

The difference between definite and indefinite integrals is that the former is a NUMBER that represents the area under the curve of a function *f(x)*, and the latter is a FUNCTION, also called an **antiderivative**, whose derivative equals the function *f(x)*. What this means is that integration can also refer to an antiderivative (indefinite integral), commonly denoted capital *F* in the form

read “the antiderivative is equal to the indefinite integral of f-of-x with respect to x”. The elongated S is the symbol for integration, where the type of integration, whether definite or indefinite, depends on whether or not there are lower and upper limits defining the domain of integration – which in this case, being an indefinite integral, there are none.

Integration involves computing the **in**definite integral (the set of all antiderivatives of the function) and evaluating at the two end-points of the **definite integral**. Calculating antiderivatives (indefinite integral) involve “reverse rules”, of which there are many. However, a common reverse rule, perhaps the most common, is called the **power rule for integration**. It goes:

where *n* equals any number you wish, and C equals the constant of integration. **Every function essentially has an infinite number of antiderivatives** the difference between them being only an additive constant. The reason for this is because the

derivative of a constant is zero, and as such, you can replace C with any constant and it’ll still be one of the infinite number of antiderivatives of that function.

You can visualize it easier by looking at it on a graph. Every antiderivative of a given function essentially has the same shape, just different vertical positions. We call this a vertical translation. By changing the value of C, the curve of the antiderivative is translated vertically – different* y*-value from other antiderivatives – but maintains the same *x*-value.

- Determining the indefinite integral is important because you can use it to find the definite integral via the fundamental theorem of calculus.

**The Fundamental Theorem of Calculus**

Developed independently by both Newton and Leibniz in the late 17th century, the fundamental theorem of calculus states that if *f* is a continuous real-valued function defined on a closed interval *[a, b]*, then, once an antiderivative *F* of *f* is known, the definite integral of *f* over that interval is given by:

You read it as *“the integral from a to b of f-of-x with respect to x is equal to the antiderivative at b minus the antiderivative at a”*. One thing to note regarding the Leibniz notation – *dy/dx* – is that they’re meant to be operators. In other words, dividing *dy/dx* isn’t necessary for the integration but simply denotes the limit of differentiation.