Finding the Equation of a Tangent Line | A Derivative-Based Approach

What is the equation of a line tangent to the curve f(x), and what is its slope?

To find the equation of a line tangent to a curve at a specific point, we can use the concept of the derivative

To find the equation of a line tangent to a curve at a specific point, we can use the concept of the derivative. The derivative of a function represents the rate of change of that function at any point on the curve.

The equation of a line tangent to the curve f(x) at a given point (x_0, f(x_0)) can be written in point-slope form as:
y – f(x_0) = m(x – x_0),

where m is the slope of the tangent line.

To find the slope of the tangent line, we need to find the derivative of the function f(x) and evaluate it at the point (x_0, f(x_0)). The derivative gives us the instantaneous rate of change of the function at that specific point.

So, the slope of the tangent line, m, is given by m = f'(x_0).

Once we have the slope, we can substitute the point (x_0, f(x_0)) into the point-slope form equation to find the equation of the tangent line. It would look like this:
y – f(x_0) = f'(x_0)(x – x_0).

To summarize:
1. Find the derivative of the function f(x).
2. Evaluate the derivative at the given point (x_0, f(x_0)) to find the slope, f'(x_0).
3. Substitute the values into the point-slope form equation: y – f(x_0) = f'(x_0)(x – x_0).

This equation represents the line tangent to the curve f(x) at the point (x_0, f(x_0)).

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