What is the equation of the normal line of #f(x)=xe^x# at #x=3 #?

Answer 1

#y-3e^3=-1/{4e^3}(x-3)#

Substituting #x=3# in the given function, we get y-coordinate of point is obtained as follows
#y=f(3)=3e^3#
Differentiating given function w.r.t. #x#, the slope of tangent #dy/dx# is
#dy/dx=f'(x)#
#=d/dx(xe^x)#
#=xe^x+e^x#
hence the slope of tangent at #x=3# is
#=f'(3)#
#=3e^3+e^3#
#=4e^3#
hence the slope #m# of normal at the same point #x=3# is given as
#m=-1/{f'(3)}#
#=-1/{4e^3}#
Now, the equation of normal at the point #(3, 3e^3)# & having slope #m=-1/{4e^3}# is given as follows
#y-3e^3=-1/{4e^3}(x-3)#
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Answer 2

The equation of the normal line of f(x)=xe^x at x=3 is y = -3e^3(x-3) + 3e^3.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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