What is the equation of the line that is normal to #f(x)= lnx^2-x# at # x= 1 #?
A normal line is simply a line perpendicular to a tangent line. We are being asked to find the normal line at
Step 1: Find the Derivative
Note first that
And that's all for this step.
Step 2: Evaluate
Here, we evaluate
But we don't want the slope of the tangent line, we want the slope of the normal line. Luckily, there is a simple relationship between tangent line and normal line slopes: they are opposite reciprocals. That is to say:
Normal line slope=-1/tangent line slope
In our case, that means the normal line slope is
Step 3: Normal Line Equation
Normal lines, like tangent lines, are of the form
Now we can solve for
The equation of the normal line is therefore