Derive the equations of tangents and normal to curves in the plane?

To derive the equations of tangents and normals to curves in the plane, we need to use the concept of derivatives.

Let's consider a curve given by the equation y = f(x), where f(x) is a differentiable function.

1. Equation of Tangent:
The equation of a tangent to the curve at a point (x0, y0) is given by the point-slope form of a line: y - y0 = m(x - x0), where m is the slope of the tangent.

To find the slope of the tangent, we take the derivative of the function f(x) with respect to x and evaluate it at x = x0. This gives us the slope of the tangent line.

So, the slope of the tangent at x = x0 is given by m = f'(x0).

Substituting this value of m in the point-slope form, we get the equation of the tangent as:
y - y0 = f'(x0)(x - x0)

2. Equation of Normal:
The equation of a normal to the curve at a point (x0, y0) is perpendicular to the tangent at that point. Therefore, the slope of the normal is the negative reciprocal of the slope of the tangent.

So, the slope of the normal at x = x0 is given by m = -1/f'(x0).

Substituting this value of m in the point-slope form, we get the equation of the normal as:
y - y0 = (-1/f'(x0))(x - x0)

These are the equations of tangents and normals to curves in the plane.