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If you are of the belief that everything has an opposite, then when it comes to the calculus you are absolutely right, in that the derivative has its very own opposite: the antiderivative. The process of taking the derivative of a function is known as differentiation. The process of going from the derivative back to the original function is known as "antidifferentiation," or integration.

Both processes have important applications in mathematics, and, by extension, the real world.
The derivative of a function f(x) is denoted many ways but most often using the symbol ' called prime as in f'(x), read "f prime of x." The derivative gives the instantaneous rate of change of the dependent variable y as the independent variable x approaches 0. To make sense out of this mumbo jumbo let us look at a specific example. Let us take the function s = -16t^2 + 60t + 5.

If you have read some of my other articles, you recognize this equation as a second-degree polynomial in t, or a quadratic; as such the graph, or picture that it produces, is that of a parabola. Moreover, since the coefficient of the t^2 term is -16, the parabola opens down.

In the above function, the independent variable is t and the dependent s, completely analogous to x and y in other functions. The reason s and t are used here is that this function gives the distance an object falls in space subject to the gravitational field of the earth, and here t represents time and s distance. Now the derivative of this function, s'(t), gives the instantaneous rate of change of s as the change in successive time intervals approaches 0. In other words, if we examine the function as time passes in ever smaller intervals-for example, as time goes from 1 second to one nanosecond later-then we are getting very close to what we mean by an instantaneous rate of change.

If we take the derivative of this distance function s we obtain s'(t) = -32t + 60. Interestingly, this turns out to be the speed or velocity of the object that the original function is describing.

That's right. The derivative of the distance function gives the velocity. For example, at t = 1 second, the original function gives that the distance fallen is 49 feet (where t is in seconds and s is in feet). You obtain this by plugging 1 into the original function. The velocity at this instant is 28 ft/sec.
Since there are times we know the funtion for velocity but not the distance, the question beomes can we find the equation which describes distance? This is where the antiderivative comes in.

For example, let us start with the velocity function, which we will call v(t) = -32t + 60. If you notice, this is the same as s'(t) above. Using the process of integration or antidifferentiation, we can find the function s(t) by finding the antiderivative of v(t) = -32t + 60. Using techniques we shall explore in another series of articles, the antiderivative turns out to be -16t^2 + 60t + c, where c stands for a constant. Now if you compare this to the original distance function you will see that c needs to be 5 in order for them to agree. So why do we get just c after we take the antiderivative of the velocity function?

Well the answer is that when we take the derivative of a constant we get 0; and when we antidifferentiate any function we must always allow for a number to be there since it vanishes when we take the derivative. This may sound a little confusing, but again in another series of articles this will become clearer. Ultimately we need to ask how we get back to that original distance function in which c = 5. The answer is that we use something called initial conditions. For example, if we are given the velocity function and told that the distance of this particle at time 0 is 5, we can then determine c to be 5.
The derivative and antiderivative are two key aspects of calculus and indeed form two of its fundamental branches: differential calculus and integral calculus. To follow will be some articles that reveal a little more about the inner workings and applications of the antiderivative. See you then.
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Joe is a prolific writer of self-help and educational material and an award-winning former teacher of both college and high school mathematics. Joe is the creator of the Wiz Kid series of math ebooks, Arithmetic Magic, the little classic on the ABC's of arithmetic, the original collection of poetry, Poems for the Mathematically Insecure, and the short but highly effective fraction troubleshooter Fractions for the Faint of Heart. The diverse genre of his writings (novel, short story, essay, script, and poetry)-particularly in regard to its educational flavor- continues to captivate readers and to earn him recognition.

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