Where Si(x) is the sine integral.
Let me explain. And to explain, I will have to introduce basically the 2 fundamental pillars of calculus. So hold on.
First, let me explain what a function is. If you know already, skip this paragraph. A function is like a machine. You feed it a number, and it comes out with another number. So, your function could be x+1. You give the number x, it gives you back x plus 1. If your function name is f, we would state this formally by saying f(x)=x+1 (The function of x is x plus 1). It just means you plug in a number, for example if you have f(2), x=2, so the function gives you back 3.
Now, a derivative is simply the rate of change of a function at a point. This just means how fast the function grows or shrinks at a certain point - you can think of it as the slope of the line that touches that point. So, let's say we are taking the derivative of the black function at the red point:
The derivative is the slope of the red line, which represents how fast the function is changing.
But you can also take the derivative of a function without knowing the point - what this means is that, let's say you are finding the derivative of f(x). You want to find a function g(x) where you can plug in a number x and it will give the derivative of f(x) at this point. Let me give an example. If we have the function f(x)=x^2, we can check by hand the derivative of f(1) is 2. And the derivative of f(2) is 4. And the derivative of f(3) is 6. Notice a pattern? The derivative of f(x) is 2x. That's finding the general derivative.
So, what is this question asking? Well, a notation you can use for the derivative of t(x) (Don't worry, the function name has changed but that doesn't affect anything - the name is arbitrary) is dt/dx. So, if we take the equation he gives us:
dt/dx * x = sin.x
And divide by x (remember your algebra! Dividing on both sides of an equation doesn't change anything, we get
Don't worry if you don't know what sin(x) is, though it's used in Hopscotch a lot you don't actually need to understand what it is to understand this. Just know it's a function.
So, what the question is saying in plain English is:
The derivative of t(x) is sin(x)/x
And it's asking us to find what t(x) is. Well, there are ways to find the derivative of a function, but how do you reverse engineer that? How do you find a function, given it's derivative? That's called the antiderivative, and it's the other pillar of calculus - integration. To integrate something is to find it's antiderivative.
Integration is the opposite of taking derivatives. Just like if you have x+2=4, you apply the opposite operator (subtraction), and you get x+2-2=4, and then the 2's cancel. It's the same. So if we integrate both sides:
Integrate ( dt/dx ) = Integrate ( sin(x)/x )
The integrate and the derivative cancel out and we are left with:
t(x) = Integrate( sin(x)/x )
Now it seems we have a problem. How can we find the antiderivative of sin(x)/x? In other words, what function is sin(x)/x the derivative of?
Try any methods you like - you won't be able to find such a function. There is no function who's derivative is sin(x)/x. I won't prove that here, but if anyone is curious I'll try to prove it in simple terms. You can put it in many forms, but for simplicity we just call the antiderivative of sin(x)/x - "Si(x)", which stands for "Sine Integral of x".
So, our final answer - if dt/dx*x=sin(x), t(x)=Si(x).