What is the equation of the normal line

Lisa Cooper

Lisa Cooper

Answered question

2022-04-09

What is the equation of the normal line of f(x)=(5+4x)2 at x=7?

Answer & Explanation

resacarno4u

resacarno4u

Beginner2022-04-10Added 12 answers

Explanation:
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 use the chain rule to differentiate (5+4x)2. By the chain rule, we know:
f(x)=2(5+4x)ddx[5+4x]
f'(x)=2(5+4x)*4
f'(x)=8(5+4x)
We can now find the slope of the tangent line at x=7.
f'(7)=8(5+4(7))=264
Now, since we want the normal line, which is perpendicular to the tangent line, we want the opposite reciprocal slope: 1264
We can use point-slope form to quickly find the equation of the normal line:
yy1=m(x{1})
We will use the point (7, 1089), which can be determined by plugging 7 into the original equation. Thus, the equation of the normal line is
y1089=1264(x7)
Aside: if you haven't yet learned the chain rule, or want another way to differentiate
(5+4x)2, just rewrite it as 25+40x+16x2, the derivative of which is
40+32x=8(5+4x)

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?