Question

Differentiating Exponential functions In Exercise,find the derivative of the function. See Example 2 and 3.

f(x) = ex + 1/ex - 1

Answer 1

`(d)/(dx)\left((e^x+1)/(e^x-1)\right)=-(2e^x)/(\left(e^x-1\right)^2)`.

Explanation:

To find the derivative of the function
`f(x)=(e^x+1)/(e^x-1)` you must:

Step 1. Apply the Quotient Rule
`\left((f)/(g)\right)'=(f\:'\cdot g-g'\cdot f)/(g^2)`

`(d)/(dx)\left((e^x+1)/(e^x-1)\right)=((d)/(dx)\left(e^x+1\right)\left(e^x-1\right)-(d)/(dx)\left(e^x-1\right)\left(e^x+1\right))/(\left(e^x-1\right)^2)`

`(d)/(dx)\left(e^x+1\right)=e^x\\\\(d)/(dx)\left(e^x-1\right)=e^x`

`=(e^x\left(e^x-1\right)-e^x\left(e^x+1\right))/(\left(e^x-1\right)^2)`

Step 2. Simplify

`\frac{{e^(2x)-e^x-e^(2x)-e^x}}{\left(e^x-1\right)^2} \\\\(-2e^x)/(\left(e^x-1\right)^2)`

Therefore,

`(d)/(dx)\left((e^x+1)/(e^x-1)\right)=-(2e^x)/(\left(e^x-1\right)^2)`