Question

How do you differentiate `f(x) = (cos2x)/(e^(2x)+4x)` using the quotient rule?

Answer 1

`f'(x) = ( -2(e^(2x)+4x)(sinx) - (2e^(2x)+4)(cos2x) ) / (e^(2x)+4x)^2`

Explanation:

If you are studying maths, then you should learn the Quotient Rule for Differentiation, and practice how to use it:

`d/dx(u/v) = (v(du)/dx-u(dv)/dx)/v^2`, or less formally, `(u/v)' = (v(du)-u(dv))/v^2`

I was taught to remember the rule in word; " vdu minus udv all over v squared ". To help with the ordering I was taught to remember the acronym, VDU as in Visual Display Unit.

So with `f(x)=(cos2x)/(e^2x+4x)` Then

`{ ("Let "u=cos2x, => , (du)/dx=-2sinx), ("And "v=e^(2x)+4x, =>, (dv)/dx=2e^(2x)+4 ) :}`

`:. d/dx(u/v) = (v(du)/dx-u(dv)/dx)/v^2`
`:. f'(x) = ( (e^(2x)+4x)(-2sinx) - (cos2x)(2e^(2x)+4) ) / (e^(2x)+4x)^2`
`:. f'(x) = ( -2(e^(2x)+4x)(sinx) - (2e^(2x)+4)(cos2x) ) / (e^(2x)+4x)^2`