How do we know the derivative of x^n

What does derivative mean? The derivative of a function the represents the gradient of the function for each value of x. Remark that the gradient of a function at point (x,f(x)) is equal to the gradient of a tangent intersecting the function at point (x,f(x)). We can use this knowledge to work out the derivative as follows: Take a secant line, intersecting the function at (x,f(x)) and (x+h,f(x+h)). Calculate gradient of secant line: dy /dx = (f(x+h)-f(x)) / ((x+h)-x) = ((x+h)ⁿ-xⁿ) / h = ((xⁿ+nx^n-1h+...+nxh^n-1+hⁿ)-xⁿ) / h = (hnx^n-1+...+hⁿ) / h = nx^n-1+h(...). As h tends to 0, the gradient of the secant line tends to that of the tangent. And at the limit h tends to 0, the term h(...) also tends to 0, so the gradient of the secant line tends to nx^n-1. So the derivative of xⁿ is nx^n-1