Question

How do you differentiate `f(x)= e^x/(e^(3-x) +2x )` using the quotient rule?

Answer 1

To differentiate a quotient, you must use the quotient rule if the function cannot be simplified.

Explanation:

To differentiate a quotient, use the following:

`(f(x)/g(x))' = (g(x)*f'(x) - f(x)*g'(x))/(g(x))^2`

To remember this you can use the "mnemonic":

"Low di hi minus hi di low all over low squared" where "di hi" or "di low" means to differentiate the top or the bottom respectively.

Using this strategy we get:

`= ((e^(3-x)+2x)(e^x)-(e^x)(-e^(3-x)+2))/(e^(3-x)+2x)^2`

Cleaning this up we can factor out an `e^x` and pull the negative from the fourth set of brackets in the numerator out in front to make a positive:

`= (e^x)[(e^(3-x)+2x)+(e^(3-x)-2))/(e^(3-x)+2x)^2`

And that's it! Hopefully this was clear and concise! Should you have any questions, please feel free to ask! :)