A Step-by-Step Guide on Finding the Equation of the Tangent Line to a Curve at a Given Point

Equation of the tangent line at a given point (a)

The equation of the tangent line to a curve at a given point (a) can be found using the concept of the derivative

The equation of the tangent line to a curve at a given point (a) can be found using the concept of the derivative. The derivative of a function represents the slope of the tangent line at any point on the graph of that function.

To find the equation of the tangent line at point (a), follow these steps:

Step 1: Find the derivative of the function. Let’s assume the function is f(x).

Step 2: Plug in the x-coordinate of the given point (a) into the derivative to find the slope of the tangent line at point (a). Let’s call this slope m.

Step 3: Use the point-slope form of a linear equation to find the equation of the tangent line. The point-slope form is y – y1 = m(x – x1), where (x1, y1) represents the coordinates of the given point. Plug in the coordinates of the point (a) and the obtained slope (m) to get the equation of the tangent line.

For example, let’s say we have the function f(x) = x^2, and we want to find the equation of the tangent line at point (a, f(a)). We need to find the derivative of f(x) first.

The derivative of f(x) = x^2 is f'(x) = 2x.

Now, plug in the x-coordinate of the given point (a) into the derivative to find the slope of the tangent line. So, m = f'(a) = 2a.

Using the point-slope form of a linear equation, the equation of the tangent line at point (a, f(a)) is: y – f(a) = 2a(x – a).

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

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