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

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.