What is the equation of the normal line of #f(x)=(5+4x)^2# at #x=7#?
The normal line to a tangent is perpendicular to the tangent at a point. Therefore, we must first find the slope of the tangent using the derivative.
We can now find the slope of the tangent line at
Now, since we want the normal line, which is perpendicular to the tangent line, we want the opposite reciprocal slope:
We can use point-slope form to quickly find the equation of the normal line:
We will use the point
Aside: if you haven't yet learned the chain rule, or want another way to differentiate