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

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resacarno4u · Accepted Answer

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&#039;(x)=2(5+4x)*4f&#039;(x)=8(5+4x)We can now find the slope of the tangent line at x=7.f&#039;(7)=8(5+4(7))=264Now, since we want the normal line, which is perpendicular to the tangent line, we want the opposite reciprocal slope: −1264We can use point-slope form to quickly find the equation of the normal line:y−y1=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 isy−1089=−1264(x−7)Aside: if you haven&#039;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 is40+32x=8(5+4x)

What is the equation of the normal line

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