Written By: Mika Lai
Introduction
When people hear the word calculus, they think of college-level mathematics, full of uninterpretable terms and symbols that the common person is incapable of comprehending. However, calculus is only viewed this way due to the new terms that -despite their lengthy spelling- are not as intimidating as one might think. Although it is safe to say that this branch of mathematics still contains its complexities, I can personally vouch that calculus is more fun, intriguing and satisfying than any other type of mathematics. So do not fear the unprecedented terms; calculus is only intimidating because it is conceptual. Once you have the concepts down, you can apply it to any question or problem that you may need to answer. This is Calculus Explained in 5 Minutes.
Differentiation
When we think of a simple graph, we see one that is linear, something similar to Figure 1. However in real life -whether it be in medicine, science or economics- graphs are non-linear; they have curves that go up and down constantly. This description is depicted by Figure 2. When we look at figure 1, the gradient is constant at any point in the graph. But in Figure 2, the gradient is not the same at every point of the graph. For example, the first slope is positive, but when it reaches the crest, the gradient is 0.
If the gradient is different at every point of the graph, is there a way to find a formula that figures out what the gradient is at any point of the graph? Meet Differentiation.
Power Rule
This formula is used to differentiate any function. Refer to the example. Take the power n, and multiply it to the constant, a. Next you subtract n by 1.
When there is a sum with multiple functions, you can apply this rule individually to each function.
Chain Rule
Developing this approach further, what if there was a function within a function? How would you differentiate that? Using the concept of the power rule, how would you differentiate the function below?
Differentiate the whole function as you would with the chain rule, bring the power down, and subtract the power by 1
Then multiply this with the differentiated function within the brackets
Note that if the function was squared, you can simply derive the function once you open the brackets instead of using the chain rule, however both ways result in the same answer! Different people just have different preferences.
Your answer should be:
Chain Rule: General Formula
Product Rule
Okay so what if you had 2 different functions multiplied together? Let’s look at the example below. Just like the others, the product rule also has a certain formula to follow, which looks like this:
Quotient Rule
On the other hand, when you have two functions or polynomials that are divided, the quotient rule can be applied. The only difference between the quotient rule and the product rule is that in the numerator you subtract, and in the quotient rule, you divide the whole thing by the function’s denominator squared. It looks like this:
Trigonometry in Calculus
Let's look at a sine graph. If we were to make another graph based off of the gradient of the sine graph at every point, what would it look like? Well if you look at Figure 3, it should reveal to you that it is identical to a cosine graph. And that is why the derivative of sin(x) is cos(x).
Does this mean that the derivative of cosine is sine? Well let’s graph it. The derivative of cos(x) is actually just -sin(x). And when you derive tan(x), you will get secant squared. Below is a chart that summarises these 3 periodic functions.
Exponentials
To end this off, here is an extremely easy rule. Exponents are exactly the same. The derivative of ex is ex.
But what if the function was e3x? Not to worry, the general formula for exponentials look like this:
So to answer the previous question, the derivative of would be 3ex.
not too hard right?
Although there are many different functions- therefore many more rules- these are the main and most frequently appearing types of functions that you will see in your equation. So let’s take this newfound knowledge to the test. Here is an example that combines the numerous rules listed above.
Solving for Minimums & Maximums
I’m sure that you have solved for minimums and maximums in quadratic equations, where you plugged the values into -b/2a to find the formula, but what happens when there are multiple maximums or minimums, such as in cubic equations, or quartic equations?
Let’s use the example below.
First we need to find the derivative, which should look like this:
Remember, the derivative is a formula for the gradient at any point of the graph. However, when we are at the minimum or maximum of a graph, the gradient is 0.
When we try to find these points all we have to do is make the derivative equal to 0, and solve for x. Make sure you solve all of the values, because there is more than one solution. In this example the answer would be negative and positive one third. This means when the x value are at these 2 coordinates, the graph will be at its minimum/maximum.
But how do we distinguish between the two? All you need to do is differentiate the derivative. We call this the second derivative. Once you have this equation you plug your x value into the graph, if y is larger than 0 the point is a minimum, and if the value is less than 0, the point is a maximum.
Conclusion
It is safe to say that calculus can be a tricky subject for many people, but one of the reasons for this is because calculus is conceptual. Once you understand the concept behind calculus, you can apply what you learned to any question. Even though calculus questions can vary, the approach is the same. So the next time you face this subject -or for any other challenging subject- don’t assume you won’t understand it, because sometimes- as shown in this case- it isn’t as hard as you think.
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