Jashan
k-12
student

hi, i am studying differentation at the moment i have drawn some graphs such as y=x ². i have found the formula for the gradient of this curve, this being 2x obtained by using differentation, but i need to know the general case for the formula where y=xⁿ  in order for me to understand this topic more throughly, i would also like to know how u derived this general formula
thankyou

Developing an expression for the differentiation of y = x ⁿ is one of the first topics that are usually covered in a beginning calculus course. In my calculus class I begin, as you did, by considering y = x ². I am not sure how you arrived at the derivative, but I take the process that I used as an introduction to the concept of a limit. We then spend some time on limits before coming back to y = x ³ and y = x ⁴ ( as well as y = x ¹). I next develop the sum, difference and product rule for diffetentiation and use the product rule to prove, by mathematical induction, that the derivative of y = x ⁿ is n x ^n-1 for n a positive integer.