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

1 Answer

#color(blue)(f' (x)=(x^2*e^(3-x)+2x*e^(3-x)+4x)/(e^(3-x)+2)^2)#

Explanation:

Start from the given #f(x)=x^2/(e^(3-x)+2)#

Use the formula #d/dx(u/v)=(v*d/dx(u)-u*d/dx(v))/v^2#

Let #u=x^2# and #v=e^(3-x)+2#

#f' (x)=d/dx(x^2/(e^(3-x)+2))#

#f' (x)=((e^(3-x)+2)*d/dx(x^2)-x^2*d/dx(e^(3-x)+2))/(e^(3-x)+2)^2#

#f' (x)=((e^(3-x)+2)*2x-x^2(e^(3-x)(0-1)+0))/(e^(3-x)+2)^2#

#f' (x)=(2xe^(3-x)+4x+x^2e^(3-x))/(e^(3-x)+2)^2#

then rearranging the numerator

#color(red)(f' (x)=(x^2e^(3-x)+2xe^(3-x)+4x)/(e^(3-x)+2)^2)#

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