Question

How do you differentiate `f(x)= ( 4- x )/ (e^x + 2)` using the quotient rule?

Answer 1

`(df(x))/(dx)=(e^x(x-5)-2)/(e^x+2)^2`.

Explanation:

The quotient rule states

`(dg(x)/(h(x)))/(dx)=((dg(x))/(dx)*h(x)-g(x)(dh(x))/(dx))/(h(x))^2`.

In this case `g(x)=4-x`, `(dg(x))/(dx)=-1`, `h(x)=e^x+2`, and `(dh(x))/(dx)=e^x`, so

`(df(x))/(dx)=((-1)(e^x+2)-(4-x)e^x)/(e^x+2)^2=(e^x(x-5)-2)/(e^x+2)^2`.