What is the equation of the normal line of #f(x)=(2+x)e^-x # at #x=0 #?
The normal line is perpendicular to the tangent line at a given point. Therefore, we must find the slope of the tangent when
We'll need to use the Product Rule.
Before we continue, we have to know what the derivatives of these functions are.
The next derivative requires use of the Chain Rule.
Let's plug our derivatives back in.
Factor out an
Now, we can compute
Now, since the normal line is perpendicular to the tangent line, its slope will be the opposite reciprocal of
In order to write the equation of the normal line, we should use point-slope form:
The point through which the normal line passes can be obtained through finding
We know that the normal line has slope
We can plug this all into point-slope form: