Bernstein polynomial derivative

Question

I am having trouble visualizing where the negative sign comes from in the final answer when differentiating the Bernstein polynomial:

$$\frac{\mathrm d}{\mathrm du}\left[B_{i,n}(u)\right] = \frac{\mathrm d}{\mathrm du}\left[{\frac{n!}{i!(n-i)!}u^i(1-u)^{n-i}}\right]$$

This simplifies to:

$$n\left[\frac{(n-1)!}{(i-1)!(n-i)!}u^{i-1}(1-u)^{n-i} + \frac{(n-1)!}{i!(n-i-1)!}u^i(1-u)^{n-i-1}\right]$$

$$\implies B_{i,n}'(u)= n[B_{i-1,n-1}(u)-B_{i,n-1}(u)]$$

From the definition of a Bernstein polynomial, where does the negative sign come from behind the second term?

Answer 1

It comes from the derivative

$$\frac{\mathrm d}{\mathrm d u}(1-u)^{n-i} = (n-i)(1-u)^{n-i-1}\frac{\mathrm d}{\mathrm du}(1-u) = - (n-i)(1-u)^{n-i-1}$$

So in the second equation, the sign should be a minus instead of a plus sign on the second part.