Understanding the Equation of a Tangent Line: Step-by-Step Breakdown and Explanation

The equation y-f(. c)=f'(. c)(x-. c) gives the equation of a line tangent to a curve at x=c

To understand this equation, let’s break it down step by step

To understand this equation, let’s break it down step by step.

First, we have the equation of a curve represented by the function f(x). The point (c, f(c)) on this curve corresponds to the value of the function at x=c.

Next, we want to find the equation of the tangent line to the curve at this point (c, f(c)). A tangent line touches the curve at only one point and has the same slope as the curve at that point.

The equation starts with y – f(c), which represents the vertical distance between any point on the tangent line (x, y) and the y-coordinate of the point (c, f(c)) on the curve. This captures how far above or below the curve the tangent line is.

The next part of the equation is f'(c), which represents the derivative of the function f(x) evaluated at x=c. The derivative of a function gives us the slope of the curve at any given point. So f'(c) gives us the slope of the curve at the point (c, f(c)).

Lastly, we have (x – c), which represents the horizontal distance between any point on the tangent line (x, y) and the x-coordinate of the point (c, f(c)) on the curve. This captures how far to the right or left the tangent line is.

Multiplying f'(c) by (x – c) gives us the change in the vertical direction caused by the horizontal distance (x – c), considering the slope of the curve at the point (c, f(c)). This term captures the slope of the tangent line and how it changes as we move horizontally away from the point (c, f(c)).

Now, adding y – f(c) and f'(c)(x – c) together gives us the equation of a line that represents the tangent line to the curve at the point (c, f(c)). It combines the vertical distance above or below the curve with the change in the vertical direction due to the slope of the curve at that point.

So, the equation y – f(c) = f'(c)(x – c) provides a way to construct the equation of a line tangent to a curve at the point (c, f(c)).

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