Hey there! You’re probably seeking a little help with calculus. You learned plenty of new symbols and functions in trig and precalc, but what is this whole dy/dx deal? We’ll explain. Note that we aren’t a math site, and a math site, your textbook, or a knowledgeable friend or teacher will generally serve you way better when it comes to mathematical queries. But hey, this blogger happens to be an engineering student. So let’s talk math.
What Does dx Mean in Calculus? Formal Definition of dx
dy/dx is a derivative.
A derivative is a tangent line that gives the slope of a graph at one particular point. As you can see in the nice graph I drew, the three colored lines are different tangent lines — that is, they are the slope of that graph at one given point.
How do you find a tangent line? In algebra, you’d estimate it with a ruler. In calc, you take the derivative.
You know how to find the slope of a line.
M = (y2 – y1) / (x2 – x1)
Or, written out, the slope is equal to the change in Y over the change in X.
Here, you want to find the slope at one point, not over the range of x1 to x2 . So you take the slope of a smaller and smaller range, until that range approaches zero. This is known as taking the limit. (If you are unfamiliar with limits, think of it as following an equation to its natural conclusion. For example, the limit of the series (½ + ¼ + ⅛ + 1/16 + 1/32 + … + ½^x) is equal to 1, because all those fractions will never add up to more than 1.
In the formal definition seen above (in blue), the slope at one point is expressed as dy/dx.
If x2 = (x1 + h), x2 – x1 = h.
Recall that y = f(x) (that is, y is a function of x). Therefore, y2 = f(x + h), and y1 = f(x).
Therefore, this equation states that the derivative, or the instantaneous slope of a line, is equal to the slope at an infinitesimally small range.
Now we’ll talk about where the notation came from, and how to calculate derivatives quickly.
What is dx in Differentiation?
dx means ‘a really small change in x.’ Slope = change in y / change in x = y/x
is a Greek letter, Delta, which in math and science means ‘change in.’ dy/dx refers to a very very small change in y and x.
Note that dy/dx is a function, not a fraction. You can’t cancel out the d’s as you would a variable.
Newton developed calculus in order to explain physics. Fun fact! The guy was also into alchemy.
What is the Difference Between dy/dx and d(fx)/dx?
So, y = f(x), or written out, y is a function of x. When you change x, y changes accordingly.
To start with, you’ll often have a function. Say f(x) = 3x2. If you were to take the derivative of that function, you would apply it to both sides of the equation, as you do any other mathematical process.
d(fx)/dx = d(3x2)/dx = 6x
How did we derive that? You can do it long-hand, using the formal definition of the derivative. But this blog is short, so we’ll explain how to use the derivative rule instead.
Now, to help you calculate derivatives faster, the derivative rule is quite handy. We recommend that you make sure you understand where derivatives come from — this blog should help, and talking to a teacher is great as well — but using the derivative rule is efficient at the moment.
f(x) = axm + c
d(fx)/dx = amxm-1
You keep any variables, a, which are multiplied or divided by x.
You take the exponent, m, down and multiply it by x.
You reduce the exponent by one.
Do this to every value of x.
Kill any constants. This one makes sense in light of the other rules (which are not bequeathed by god, we’re just limited in wordcount. Your textbook should explain).
f(x) = a(xm) + c(x0) [x0 = 1]
d(fx)/dx = amxm-1 + c*0*x-1 = amxm-1 + 0
Again, we urge you to consult a teacher, textbook, or friend for further understanding of this.
Why do Integrals Always Have a dx?
We’re not going to explain integrals here, but basically, finding the integral involves finding the reverse of a derivative. The integral of amxm-1 , for example, is equal to a(xm) + c.
d(fx)/dx = amxm-1
You need to integrate over both sides, so first you treat d(fx)/dx as a fraction and multiply both sides of the equation by dx.
d(fx) = (amxm-1)(dx)
then you can integrate over both sides, and the integral will eat the d. Yes. We’re literally writing that, as if it’s all magic. It’s not. Einstein said that you don’t understand something thoroughly until you can explain it to a five-year-old, and frankly?
Okay, moving on.
∫ d(fx) = ∫ (amxm-1)(dx) ∫ is the notation for integral
∫ d(fx) = fx = a(xm) + c
Integration and derivation are opposite functions, like addition and subtraction, so one kinda undoes the other. Also, this makes the derivative disappear.
You can integrate to find a(xm) + c without knowing the original function fx, but that falls to websites more advanced than ours. Again, we recommend calling on the wisdom of teachers and textbooks.
Wrapping Things Up: What dx Means in Calculus
You just had a little calculus crash course, and hopefully you are not more confused now than you were before. In time and with better, more thorough explanations, it will all make sense.
To summarize, dx is a really small change in x. You use derivatives to find the instantaneous slope of a line, which is to say, the change in y over a really really small change in x.
Taking the derivative of an equation is a function, such as sin(x) or log(x), and must be done to both sides of the equation.
Integrals are the opposite of derivatives. They are snarly and confusing, but in simple form, they ‘undo’ the derivative and go back to the original function.
Also, another way to write dy/dx is simply f’(x) (read ‘f prime’).
Hope this helped!
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