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” Knowing why something is changing at an exact moment is the heart of calculus. For example, calculus tells you not only the speed of your car, but how much that speed is changing at any given moment. This is one of the simplest uses of calculus, but it is incredibly important. Imagine how useful that knowledge would be for the speed of a spaceship trying to get to the moon!  Finding instantaneous change is called differentiation. Differential calculus is the first of two major branches of calculus. A "derivative" is a fancy sounding word that inspires anxiety. The concept itself, however, isn't that hard to grasp -- it just means "how fast is something changing." The most common derivatives in everyday life relate to speed. You likely don’t call it the “derivative of speed,” however – you call it "acceleration." Acceleration is a derivative – it tells you how fast something is speeding up or slowing down, or how the speed is changing. This is one of the key findings of calculus. The rate of change between two points is equal to the slope of the line connecting them. Think of a basic line, such as the equation y=3x.{\displaystyle y=3x.} The slope of the line is 3, meaning that for every new value of x,{\displaystyle x,} y{\displaystyle y} changes by 3. The slope is the same thing as the rate of change: a slope of three means that the line is changing by 3 for every change in x.{\displaystyle x.} When x=2,y=6;{\displaystyle x=2,y=6;} when x=3,y=9.{\displaystyle x=3,y=9.}  The slope of a line is the change in y divided by the change in x.  The bigger the slope, the steeper a line. Steep lines can be said to change very quickly. Review how to find the slope of a line if your memory is hazy. Finding the slope of a straight line is relatively straightforward: how much does y{\displaystyle y} change for each value of x?{\displaystyle x?} Yet complex equations with curves, like y=x2{\displaystyle y=x^{2}} are much harder to find. However, you can still find the rate of change between any two points – simply draw a line between them and calculate the slope. For example, in y=x2,{\displaystyle y=x^{2},} you can take any two points and get the slope. Take (1,1){\displaystyle (1,1)} and (2,4).{\displaystyle (2,4).} The slope between them would equal 4−12−1=31=3.{\displaystyle {\frac {4-1}{2-1}}={\frac {3}{1}}=3.} This means that the rate of change between x=1{\displaystyle x=1} and x=2{\displaystyle x=2} is 3. The closer your two points, the more accurate your answer. Say you want to know how much your car accelerates right when you step on the gas. You don’t want to measure the change in speed between your house and the grocery store, you want to measure the change in speed the second after you hit the gas. The closer your measurement is to that split-second moment, the more accurate your reading will be. For example, scientists study how quickly some species are going extinct to try to save them. However, more animals often die in the winter than the summer, so studying the rate of change across the entire year is not as useful – they would find the rate of change between closer points, like from July 1st to August 1st. This is where calculus often becomes confusing, but this is actually the result of two simple facts.  First, you know that the slope of a line equals how quickly it is changing. Second, you know that closer the points of your line are, the more accurate the reading will be. But how can you find the rate of change at one point if slope is the relationship of two points? The answer: you pick two points infinitely close to one another. Think of the example where you keep dividing 1 by 2 over and over again, getting 1/2, 1/4, 1/8, etc. Eventually you get so close to zero, the answer is "practically zero." Here, your points get so close together, they are "practically instantaneous." This is the nature of derivatives. There are a lot of different techniques to find a derivative depending on the equation, but most of them make sense if you remember the basic principles of derivatives outlined above. All derivatives are is a way to find the slope of your "infinitely small" line. Now that your know the theory of derivatives, a large part of the work is finding the answers. Using derivatives to find the rate of change at one point is helpful, but the beauty of calculus is that it allows you to create a new model for every function. The derivative of y=x2,{\displaystyle y=x^{2},} for example, is y′=2x.{\displaystyle y^{\prime }=2x.} This means that you can find the derivative for every point on the graph y=x2{\displaystyle y=x^{2}} simply by plugging it into the derivative. At the point (2,4),{\displaystyle (2,4),} where x=2,{\displaystyle x=2,} the derivative is 4, since y′=2(2).{\displaystyle y^{\prime }=2(2).}  There are different notations for derivatives. In the previous step, derivatives were labeled with a prime symbol – for the derivative of y,{\displaystyle y,} you would write y′.{\displaystyle y^{\prime }.} This is called Lagrange's notation. There is also another popular way of writing derivatives. Instead of using a prime symbol, you write ddx.{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}.} Remember that the function y=x2{\displaystyle y=x^{2}} depends on the variable x.{\displaystyle x.} Then, we write the derivative as dydx{\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} x}}} – the derivative of y{\displaystyle y} with respect to x.{\displaystyle x.} This is called Leibniz's notation. The easiest example is based on speed, which offers a lot of different derivatives that we see every day. Remember, a derivative is a measure of how fast something is changing. Think of a basic experiment. You are rolling a marble on a table, and you measure both how far it moves each time and how fast it moves. Now imagine that the rolling marble is tracing a line on a graph – you use derivatives to measure the instantaneous changes at any point on that line.  How fast does the marble change location? What is the rate of change, or derivative, of the marble’s movement? This derivative is what we call “speed.” Roll the marble down an incline and see how fast in gains speed. What is the rate of change, or derivative, of the marble’s speed? This derivative is what we call “acceleration.” Roll the marble along an up and down track like a roller coaster. How fast is the marble gaining speed down the hills, and how fast is it losing speed going up hills? How fast is the marble moving exactly halfway up the first hill? This would be the instantaneous rate of change, or derivative, of that marble at its one specific point.

Summary:
Know that calculus is used to study “instantaneous change. Use derivatives to understand how things change instantaneously. Know that the rate of change is the slope between two points. Know that you can find the slope of curved lines. Make your points closer together for a more accurate rate of change. Use infinitely small lines to find the “instantaneous rate of change,” or the derivative. Learn how to take a variety of derivatives. Find derivative equations to predict the rate of change at any point. Remember real-life examples of derivatives if you are still struggling to understand.