How come reducing the power of a function by 1 and multiplying the function by it gives us derivative of the function?

Question

Is it just a coincidence, or is it like an assumption and everything else is built around it?

In: 0

Answer 1

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.