Question

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

Answer 1

`(df)/(dx)=(-e^x(2x^2-7x+3)-8x+6)/(-e^x+2)^2`

Explanation:

Quotient rule states that if `f(x)=g(x)/(h(x))`, then

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

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

`(df)/(dx)=((-e^x+2)(-4x+3)-(-e^x)(-2x^2+3x))/(-e^x+2)^2`

= `(4xe^x-3e^x-8x+6-2x^2e^x+3xe^x)/(-e^x+2)^2` or

= `(-e^x(2x^2-7x+3)-8x+6)/(-e^x+2)^2`