Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

$$\dfrac{1}{1+e^{-x}} = \dfrac{e^x}{1+e^x}$$

I was told to sketch a curve but couldn't figure out the first step. The solution manual rewrote the left hand side of the equation above as the right hand side. I cannot figure out what they did to get this. Could someone explain this to me?

share|cite|improve this question
Multiply top and bottom of the lhs by $e^x$. About your title: the two functions are equal, which yields one equation. – 1015 May 24 '13 at 0:12
Try multiplying the top and bottom by $e^{x}$. – Andy Bromberg May 24 '13 at 0:12
up vote 6 down vote accepted

$$\frac{1}{1+e^{-x}}=\frac{e^x}{e^x}\frac{1}{1+e^{-x}}=\frac{e^x}{e^x+e^x\cdot e^{-x}}=\frac{e^x}{e^x+1}$$

share|cite|improve this answer

They just multiplied the fraction by $1$, in the form $\dfrac{e^x}{e^x}$.

share|cite|improve this answer

Multiply the left hand side by $1 = e^x/e^x$. So, you have

$$\frac{1}{1+e^{-x}}\cdot\frac{e^x}{e^x} = \frac{e^x}{1+e^{x}}$$ $$\frac{e^x}{e^x + e^{x-x}} = \frac{e^x}{1+e^{x}}$$ $$\frac{e^x}{e^x + 1} = \frac{e^x}{1+e^{x}}$$

($e^0 = 1$)

share|cite|improve this answer

Left-hand side of the equation is

$$\dfrac{1}{1+e^{-x}} = \dfrac{1}{1 +\dfrac{1}{e^{x}}}=\dfrac{1}{\dfrac{e^{x} +1}{e^x}}=\dfrac{e^x}{e^x+1},$$

which is equal to the right-hand side $\dfrac{e^x}{1+e^x}$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.