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

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).