Calculus Definitions > Differentiate Definition
What Does Differentiate Mean?
When you differentiate (or “take the derivative”), you’re finding the slope of a function at a particular point. It tells you the rate of change (i.e. how fast or how slow something is changing). For example, if you know the position of a car, you can use differentiation to tell you how fast the car is going at that point.
Let’s say you have a function that models where a car is at any particular point in time. If you differentiate that position function, you end up with a velocity function. In other words, you start with one piece of information (position) and you end up with two pieces of information (position and velocity). In fact, if you know the position function of any object, you can use that to find acceleration and “jerk” as well (jerk is that stomach churning feeling you get when an elevator makes sudden stops and starts). So, differentiating a function allows you to extract a lot of useful information from just a little bit of knowledge.
Slope Formula vs Differentiate Definition
At this point, you may be wondering why you can’t use the slope formula from algebra to find the slope. The answer is: you still can. There’s no difference between finding a slope using the slope formula, or finding the slope using differentiation. The major difference is that differentiation can give you all of the slopes at all of the points, while the slope formula can only give you one at a time. Let’s say you had a thousand coordinate points for the position of a new planet. You can either plug those in a thousand times to the slope formula or you can differentiate once. It’s faster, easier, and a lot less tedious.
Differentiation Definition: Rules
Many rules for differentiation have been formulated. They turn the chore of calculating a zillion slopes or working with the equally tedious limit formulas into relatively quick formulas. You can use the rules to find derivatives quickly for many different types of functions. For example, use the quotient rule if you have two functions being divided, like (x2) / (x + 3x) and use the product rule if they are multiplied together instead.
Watch this short video for an example of how the simple to use power rule works to differentiate power functions:
Sometimes it’s difficult, or impossible to solve an equation for x. For example, complicated functions like 2y2 -cos y = x2 cannot easily be solved for x. In these cases, you’ll have to use a more advanced method called implicit differentiation.
Differentiate Definition References
Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, 1999
Beyer, W. H. “Derivatives.” CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 229-232, 1987.
Stephanie Glen. "Differentiate Definition (“Take the Derivative”)" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/differentiate-definition/
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