Is it just a coincidence, or is it like an assumption and everything else is built around it?
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It’s a coincidence that happens to work for polynomials.
It wasn’t an assumption. It starts with the definition of a derivative. When you work out the math, that’s what the answer turns out to be.
Take the function f(x) = x^2. We can rewrite it as f(x) = x * x. The derivative of a function is the slope of the line tangent to the graph of the function at a given point. It tells us how fast the function is changing at that given point. Now, if we take away 1 from x^2, we are really just taking away 1 from one of the factors in our rewritten equation, so we are left with (x)(x – 1). The slope of this new equation at any given point will be different than the slope of our original equation, because we’ve changed one of the factors. So, when we reduce a power by 1 and multiply by the original function, we are essentially taking its derivative.
In mathematics concepts can be defined in many ways: we like to talk about equivalent characterizations. For example the exponential function has [6 equivalent characterizations](https://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function). Note that the derivative of the exponential function is itself hence your assumption doesn’t hold here.
I like to think of the reverse process. An integral is the inverse of a derivative. The derivative gives us the slope of the tangent line. The integral gives us the area underneath the curve.
A flat, horizontal line would be described as y=c let’s just say its 5 units up, so y=5. If you want to know the area under this line, whatever region you pick is going to be a rectangle. The height is 5 and the width is the distance along the x-axis. So the integral is 5x.
A slanted line would be described as y=mx+c. Let’s just say the line has a slope of 4 and intersects the origin. The line is y=4x and the area under this curve looks like a triangle where the height is the value for y (we know y=4x so the height is 4x) and the base is the distance along the x axis. So the area of a triangle: 1/2*b*h becomes 1/2*x*4x or 2x²
If the inverse of the integral is the derivative, then we just work backwards to find the derivative. The derivative of 2x² must be 4x. Likewise, the derivative of 5x must be 5. The area of a parabola is not quite so easy but you already know how it’s going to turn out from the trick we use. The easiest way to find out, though, is to split up the function into thinner and thinner rectangles and just add them all up. You’ll see that the answer gets closer and closer to the same pattern.
Something of a nitpick: This doesn’t apply to all functions, only functions of the form ax^b where b isn’t 0. As others have said, you can show that it is the case from the definition of a derivative, but it is not itself really anything special.
Another way to see this is as an application of the product rule and the slope of a line.
We already know from simple geometry that the slope (at any point) of a function like f(x)=ax (where a is constant) is just f'(x)=a.
Now just treat x^(3) as x*x*x and apply the product rule, which instructs us to take the derivative of each part separately while holding the other parts constant. This gives:
x*x + x*x + x*x = 3x^(2)
Then picture this process for any arbitrary exponent, and the same pattern holds. In each of the individual product rule pieces, the number of ‘x’s decreases by 1 because we’re taking a derivative, and the number of pieces the number of ‘x’s we started with.
What if you had f(x) = x^n, and you didn’t know the formula for it (reducing the power by 1 and multiplying)?
You can still find the derivative. You just have to go from the definition of derivative.
– The derivative is defined as f'(x) = lim d->0 of (f(x+d) – f(x)) / d.
– For our f(x) = x^n, this gives you f'(x) = ((x+d)^n – x^n) / d.
– If you expand out with binomial theorem, you get (x^n + ndx^(n-1) + … + d^n – x^n) / d.
– Simplifying, the x^n cancels, then you divide the ndx^(n-1) + … + d^n part by d.
– This gives you nx^(n-1) + … + d^(n-1), but every term in the … part is multiplied by d, so it vanishes as d goes to 0, leaving you with nx^(n-1).
No, it is not a coincidence and not an assumption. It is a consequence of a fact, that “x^(n)” is a “volume” of an n-dimensional “cube”.
Derivative answers the question: how much does a function grow, if we make “x” slightly larger? In case of “x^(n)”, it means: how much does a volume of nD-cube grows, when we make side length slightly larger?
* The nD-cube has “n” directions to grow
* The “grow border” in each direction is an (n-1)-dimensional cube.
So, the total growing rate is equal (number of directions)*(border size in each direction) = n*x^(n-1)
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