Question

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

Answer 1

`color(red)(f' (x)=((e^x*csc x*cot x+2*csc x*cot x -4*e^x+e^x*csc x))/((e^x+2)^2))`

Explanation:

Start from the given function

`f(x)=(4-csc x)/(e^x+2)`

The formula to be used in finding the derivative is

`d/dx(u/v)=(v d/dxu-ud/dxv)/v^2`

Let `u=(4-csc x)` and `v=(e^x+2)`

`f' (x)=d/dx((4-csc x)/(e^x+2))=((e^x+2) d/dx(4-csc x)-(4-csc x)d/dx(e^x+2))/((e^x+2)^2)`

`f' (x)=((e^x+2) (0-(-csc x*cot x))-(4-csc x)(e^x*1+0))/((e^x+2)^2)`

`color(red)(f' (x)=((e^x*csc x*cot x+2*csc x*cot x -4*e^x+e^x*csc x))/((e^x+2)^2))`

God bless....I hope the explanation is useful.