Question

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

Answer 1

The quotient rule is that, if you have `f(x)=g(x)/(h(x))`, then `f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x)^2)`.

Here, we will take `g(x)=cos^2(x)`, and `h(x)=e^(x-2)+x`

`g'(x)=-2sinxcosx`

`h'(x)=e^(x-2)+1`

So, `f'(x)=((e^(x-2)+x)(-2sinxcosx)-(e^(x+2)+1)(cos^2x))/(e^(x+2)+x)^2=-(cosx(e^(x-2)+x)(2sinx)-(e^(x+2)+1)(cosx))/(e^(x+2)+x)^2`