The Equation of a Tangent and Normal at any Point on a Curve
The Equation of a Tangent and Normal at any Point on a Curve
The Concept of Tangents and Normals

Tangent at a point on the curve is a straight line that just touches the curve at that point. This line has the same gradient as the curve at that point.

Normal at the point on the curve is a straight line that is perpendicular to the tangent at that point.
Deriving the Equation of a Tangent

Step 1: Find the first derivative of the given function. This derivative is the gradient function.

Step 2: Substitute the xvalue of the given point into the first derivative to find the gradient of the tangent.

Step 3: Use the gradientpoint form of a line equation (y  y1 = m(x  x1)) where m is the gradient and (x1, y1) is the given point, to determine the equation of the tangent.
Deriving the Equation of a Normal

Step 1: Find the gradient of the tangent at the point (as explained above).

Step 2: Calculate the negative reciprocal of the gradient of the tangent to find the gradient of the normal (m_tangent * m_normal = 1).

Step 3: Again, use the gradientpoint form of a line (y  y1 = m(x  x1)), but this time, use the gradient of the normal and the given point to gain the equation of the normal.
Remember, always round off final answers to an appropriate degree of accuracy, especially when gradients are irrational or comprise of decimals.