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+nxn-1h+...+nxhn-1+hn)-xn) / h = (hnxn-1+...+hn) / h = nxn-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 nxn-1. So the derivative of xn is nxn-1

Answered by Thomas P. Maths tutor

2835 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

Use logarithms to solve the equation 3^(2x+1) = 4^100


Differentiate 3x^(3) + 7x^(2) -4x


differentiate 3x^56


f(x) is defined by f(x) = 3*x^3 + 2*x^2 - 7*x + 2. Find f(1).


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences