Equation of The Tangent

One of the most prominent applications of differentiation is the “tangent line.” Tangent is derived from the Latin word “tangere,” which means “to touch.” At one point on the curve, the tangent line hits it. To calculate the tangent line equation, we must first determine the curve’s equation (which is given by a function) and the location where the tangent is drawn. The “point of tangency” is the location where the tangent is drawn.

Tangent lines 

At a given point, the tangent line of a curve is a line that just touches the curve (function). In calculus, the tangent line may contact the curve at any other point(s) and cross the graph at any other point(s). It is NOT a tangent line of the curve at each of the two points if a line travels through two points of the curve but does not contact the curve at either of the places. The line is known as a secant line in this circumstance. We may see several examples of tangent and secant lines in this diagram. 

Tangent to a curve

The intuitive sense that a tangent line “touches” a curve can be clarified by looking at the succession of straight lines (secant lines) that pass through two points on the function curve, A and B. When point B approaches or tends to A, the tangent at A is the limit. The tangent line’s existence and uniqueness are dependent on a sort of mathematical smoothness known as “differentiability.” If two circular arcs intersect at a sharp point (a vertex), for example, there is no uniquely defined tangent at the junction because the progression of secant lines is limited by the direction “point B” approaches the vertex.

Slope of tangent line

Consider the curve f, which is represented by a function (x). Consider a secant line that passes through two points on the curves P (x₀, f(x₀)) and Q (x₀ + h, f(x₀ + h) respectively. P and Q are at a distance of h units from one another.

Using the slope formula, the slope of the secant line is,

Secant line slope = [f(x₀ + h) – f(x₀)] / [f(x₀ + h) – f(x₀)] = (x₀ + h) / h

The secant line becomes the tangent line at P if Q reaches very close to P (by making h₀) and merges with P, as shown in the diagram above. H₀ can be applied to the slope of the secant line to get the slope of the tangent line at P. So

Tangent Slope at P = lim(h -> 0 ) [f(x₀ + h) – f(x₀)] / h

By the limit definition of the derivative (or) basic principles), we know that this is nothing but the derivative of f(x) at x = x₀. 

As a result, the tangent’s slope is simply the derivative of the function at the place where it is drawn.

Tangent Line Slope Formula

(dy/dx) or (f ‘(x)) is the slope of the tangent line of y = f(x) at a point (x₀, y₀), where

The derivative of the function f is f’(x) .

The value produced by replacing (x, y) = (x₀, y₀) in the derivative f ‘(x) is (x₀, y₀).

If the function is implicitly defined, we may have to utilise implicit differentiation to obtain the derivative f’(x).

Tangent line approximation

The concept of linear approximation is derived directly from the tangent line equation. Specifically, the equation of the tangent line of a function y = f(x) at a point (x₀, y₀) can be used to approximate the value of the function at any point near (x₀, y₀). The tangent line to a point on a differentiable curve is also known as a tangent line approximation, which is the graph of the affine function that best approximates the original function at a given point.

Conclusion

Therefore we can finally conclude that the slope of a horizontal tangent is 0 since it is parallel to the x-axis. We know that the slope is equal to the function’s derivative. Set the derivative of the function to zero and solve to discover the points where there are horizontal tangents. We can use the point-slope form to calculate the equation of the horizontal tangent line after we have the points.