Given #f(x)=-x/(e^(x-x^3))# at # x=-1#
Determine the first derivative #f' (x)# first, then compute for #f' (-1)# to obtain the slope
We make use of the derivative of quotient for this type of function
use the formula #d/dx(u/v)=(v*d/dx(u)-u*d/dx(v))/v^2#
Let #u=x# and #v=e^(x-x^3)#
#f' (x)=d/dx(f(x))=#
#d/dx(-x/(e^(x-x^3)))=(-1)*(e^(x-x^3)*d/dx(x)-x*d/dx(e^(x-x^3)))/(e^(x-x^3))^2#
#d/dx(-x/(e^(x-x^3)))=(-1)*((e^(x-x^3)*1-xe^(x-x^3)(1-3x^2)))/(e^(x-x^3))^2#
There is no need for further simplification just use #x=-1# right away
#f' (-1)=(-1)*((e^(-1-(-1)^3)*1-(-1)e^(-1-(-1)^3)(1-3(-1)^2)))/(e^(-1-(-1)^3))^2#
#f' (-1)=(-1)*((e^(-1+1)-(-1)e^(-1+1)(1-3)))/(e^(-1+1))^2#
#f' (-1)=(-1)*((e^(0)-(-1)e^(0)(-2)))/(e^(0))^2#
#f' (-1)=(-1)*((1-2))/(1)^2#
#f' (-1)=+1#
God bless....I hope the explanation is useful.
