How To Find the Normal to a Curve at x = a | Formula Proof: h(x) = (−1/f′(a))(x − a) + f(a)

How To Find the Normal to a Curve at x = a | Visual Calculus Proof

In this video, we derive the equation of the normal to a curve at the point (a, f(a)) using derivatives, perpendicular slopes, and the geometry of tangent lines.

For the curve:

y = f(x)

when x = a, the point on the curve is:

(a, f(a))

The slope of the tangent line is:

f′(a)

Since perpendicular slopes satisfy:

m₁m₂ = −1

the slope of the normal line is:

−1/f′(a), f′(a) ≠ 0

Using the equation of a line:

y = mx + c

we derive the formula for the normal line:

h(x) = (−1/f′(a))(x − a) + f(a)

This video visually explains:
• Derivatives as slopes
• Tangent and normal lines
• Perpendicular gradients
• Geometric intuition in calculus
• The derivation of the normal line formula

Follow Mathematics Proofs:

𝕏 (Twitter):
https://www.x.com/tiago_hands

Instagram:
https://www.instagram.com/mathematics.proofs/

Threads:
https://www.threads.net/@mathematics.proofs

TikTok:
https://www.tiktok.com/@mathematics.proofs

Amazon Store:
https://www.amazon.co.uk/shop/mathematics.proofs

Видео How To Find the Normal to a Curve at x = a | Formula Proof: h(x) = (−1/f′(a))(x − a) + f(a) канала Mathematics Proofs - GCSE & A Level