What is the equation of the line that is normal to #f(x)= x-e^sqrt(x+4)/(x-4) # at # x= 5 #?

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Answer 1 · 2017-12-25T10:41:00

#y=5−e^3-6/(6+5e^3)*(x-5)#

#f(x)=x−(e^(sqrt(x+4)))/(x−4);quadquadquad# #a=5#

#y=f(a)-1/(f'(a))*(x-a)#

#f'(x)=1-(e^(sqrt(x+4))1/2(x+4)^(-1/2)(x-4)-e^(sqrt(x+4)))/(x-4)^2#

#f'(5)=1-(e^(sqrt(5+4))1/2(5+4)^(-1/2)(5-4)-e^(sqrt(5+4)))/(5-4)^2#

#f'(5)=1-(e^3(1/2)(9)^(-1/2)-e^3)/1=1-(e^3(1/(2*3))-e^3)#

#f'(5)=1-e^3(1/6-1)=1-e^3(-5/6)=1+5/6e^3=(6+5e^3)/6~~17.7379#

#f(5)=5−(e^(sqrt(5+4)))/(5−4)=5−e^3~~-15.085#

#y=5−e^3-6/(6+5e^3)*(x-5)#