## Equation of the tangent at X=N

### The equation of a tangent line at a given point on a curve can be determined using the derivative of the function representing the curve

The equation of a tangent line at a given point on a curve can be determined using the derivative of the function representing the curve. The process involves finding the slope of the tangent line at the given point and then using the point-slope form of a line to write the equation.

To find the equation of the tangent line at point X = N, follow these steps:

1. Determine the equation of the curve or function. Let’s say the equation of the curve is given by f(x).

2. Calculate the derivative of f(x) using the appropriate differentiation rules. This will give you the slope of the curve at any point.

d/dx[f(x)] = f'(x)

3. Substitute X = N into the derivative obtained in step 2 to find the slope of the tangent line at the given point.

Slope of tangent line = f'(N)

4. Use the point-slope form of a line to write the equation of the tangent line. Since we have the slope from step 3 and the point (X = N, Y = f(N)) on the curve, we can write the equation as:

y – f(N) = f'(N)(x – N)

This is the equation of the tangent line at X = N.

Note: In some cases, the derivative f'(N) might equal zero, indicating a horizontal tangent. In such cases, the equation of the tangent line becomes:

y = f(N)

Definitions

1. Function: A function is a mathematical relationship between an input (usually denoted as x) and an output (usually denoted as y or f(x)). It assigns exactly one output value to each input value. For example, y = 2x + 1 is a linear function.

2. Derivative: The derivative of a function represents the rate at which the function is changing at each point. It measures the slope of a curve at any given point. The derivative of a function f(x) is denoted as f'(x) or df/dx.

3. Slope: Slope measures the steepness or incline of a line. It determines how much a line rises or falls as the x-coordinate increases. Slope is denoted as m and is calculated as the change in Y-coordinates divided by the change in X-coordinates (rise over run).

4. Point-slope form: The point-slope form of a linear equation represents the line passing through a specific point (x₁, y₁) with a given slope (m). It is written as y – y₁ = m(x – x₁). This form is often used to write the equation of a tangent line, where the point is a known coordinate on the curve and the slope is obtained from the derivative.

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