We start from the given equation #f(x)=cos x-e^x sin x#
Let us solve for the point of tangency first
#f(pi/3)=cos (pi/3)-e^(pi/3) sin (pi/3)#
#f(pi/3)=1/2-e^(pi/3) sqrt(3)/2#
Let us solve for the slope #m# now
#f(x)=cos x-e^x sin x#
Find the first derivative first
#f' (x)=d/dx(cos x-e^x sin x)#
#f' (x)=-sin x-[e^x*cos x+sin x*e^x*1]#
Slope #m=f' (pi/3)=-sin (pi/3)-[e^(pi/3) cos (pi/3)+sin (pi/3)*e^(pi/3)]#
#m=f' (pi/3)=-sqrt(3)/2-[e^(pi/3) *1/2+sqrt(3)/2*e^(pi/3)]#
#m=f' (pi/3)=-sqrt(3)/2-[1/2+sqrt(3)/2]*e^(pi/3)#
#m=f' (pi/3)=-1/2[sqrt(3)+e^(pi/3)+sqrt(3)e^(pi/3)]*#
Our Tangent Line:
#y-f(pi/3)=m(x-pi/3)#
#y-1/2+ sqrt(3)/2*e^(pi/3)=-1/2(sqrt(3)+e^(pi/3)+sqrt(3)e^(pi/3))(x-pi/3)#
Kindly see the graph of #f(x)= cos x-e^x sin x# and the tangent line
#y-1/2+ sqrt(3)/2*e^(pi/3)=-1/2(sqrt(3)+e^(pi/3)+sqrt(3)e^(pi/3))(x-pi/3)#
God bless....I hope the explanation is useful.
