How do I go from the first expression to the second?

Question

Induction proof $$\begin{align} = & (k-1)!(-1)^{k-1}(-kx^{-k-1})\\ = & k!(-1)^k x^{-(k+1)} \end{align}$$

I simply don't understand how they get from the first line to the second one. Could someone explain that to me?

Answer 1

It is $$(k-1)!\times(-k)\times(-1)^{k-1}x^{-(k-1)}=k!\times(-1)^kx^{-(k-1)}$$ Using this $$(k-1)!k=k!$$ $$(-1)\times(-1)^{k-1}=(-1)^k$$

Answer 2

Well, just rearrange the product and use the definition of forming powers:

$(k-1)! (-1)^{k-1}(-kx^{-k-1}) = [(k-1)!\cdot k]\cdot [(-1)^{k-1}\cdot(-1)]\cdot x^{-k-1} = k!\cdot (-1)^k\cdot x^{-(k+1)}.$

Answer 3

In the factor $-k$, $k$ was grouped with $(k-1)!$ (whence $k!$) and the $-$ sign was grouped with $(-1)^{k-1}$ (whence $(-1)^k$).