f(x)=x^n,求f'(x)lim(dx->0)[C(n,1)*x^(n-1)*dx+C(n,2)*x^(n-2)*(dx)^2+...+(dx)^n]/dx是怎么化解成nx^(n-1)的

Question

f(x)=x^n,求f'(x) lim(dx->0)[C(n,1)*x^(n-1)*dx+C(n,2)*x^(n-2)*(dx)^2+...+(dx)^n]/dx
是怎么化解成nx^(n-1)的

Answer 1

lim(dx->0)[C(n,1)*x^(n-1)*dx+C(n,2)*x^(n-2)*(dx)^2+...+(dx)^n]/dx
consider
(x+dx)^n
x^n + C(n,1)x^(n-1) dx +.+ (dx)^n
lim(dx->0)[C(n,1)*x^(n-1)*dx+C(n,2)*x^(n-2)*(dx)^2+...+(dx)^n]/dx
= lim(dx->0) [(x+dx)^n - x^n] /dx (0/0)
=lim(dx->0) n(x+dx)^(n-1)
=nx^(n-1)