Chain rule for derivatives and simplification question.

Question

I am trying to find the derivative of the function $$-\frac 13 (e^x −1)^2 +(e ^x−1)+ \frac 15$$ I can only get as far as:

$$e^x - \frac{ 2(e^x - 1)e^x}{3}.$$

The answer I am looking for is $$-\frac13 (e^x(2e^x-5)).$$ For the life of me I can't see how it is being simplified this way, especially where that -5 is coming from. Any help would be awesome!

Answer 1

$$\begin{align} e^x - \frac{ 2(e^x - 1)e^x}{3}&=e^x\left(1-\frac{2(e^x-1)}{3}\right)\\ &=e^x\left(\frac{3-2e^x+2}{3}\right)\\ &=-\frac{1}{3}e^x(2e^x-5) \end{align}$$

Answer 2

Hint:

From where you got, you can factor out $e^x$: $$\to e^x[1-\frac 23 (e^x-1)]$$