The top is just the binomial expansion of
[x + (1 - x)]^n = 1^n = 1
The derivative of 1 is obviously 0, so the right hand side of the bottom is correct. As for the left-hand side, the binomial coefficients C(n, k) are just constants and don't get changed by differentiation, so the critical part is differentiating
x^k(1 - x)^(n - k)
product rule:
kx^(k - 1)(1 - x)^(n - k) + x^k[(1 - x)^(n - k)]'
chain rule on second term:
kx^(k - 1)(1 - x)^(n - k) + x^k(n - k)(1 - x)^(n - k - 1)(-1)
kx^(k - 1)(1 - x)^(n - k) - x^k(n - k)(1 - x)^(n - k - 1)
take out common factor of x^(k - 1)(1 - x)^(n - k - 1):
x^(k - 1)(1 - x)^(n - k - 1)[k(1 - x) - x(n - k)]
rewrite in given form:
[k(1 - x) - (n - k)x]x^(k - 1)(1 - x)^(n - k - 1)
