You are trying to prove by induction that . So you assume the result for n=k, and you are trying to prove it for n=k+1 by differentiating one more time. Where you are going wrong is that you are finding the (k+1)th derivative of the same function. But the result for n=k+1 refers to the (k+1)th derivative of the function . You can differentiate that k+1 times by regarding it as a product (as I just indicated) and using Leibniz's rule together with the inductive hypothesis.
so i made the k+1 derivative and i need to multiply both sides by the n+1 derivative of x
in order to make it the k+1 expression
so i need to do the derivative of x*f(x)=1*f(x)+x*f'(x)
of the left side
i have both f(x) and f'(x)
so if i will substitute
it will work??