How to Find the Slope of a Tangent Line

The slope of a tangent line to a curve at a specific point is determined by finding the derivative of the function describing the curve and evaluating it at the desired point.

Steps to Find the Slope of a Tangent Line

1. Understand the Function

Let the curve be represented by f(x), where f(x) is a differentiable function.

2. Find the Derivative

The derivative f′(x)f'(x)f′(x) represents the slope of the tangent line at any point $x$ on the curve.

3. Evaluate the Derivative at the Given Point

To find the slope at a specific point (x0, f(x0), substitute x0x_0x0​ into the derivative:

Slope=f′(x0​)

Example 1: Polynomial Function

Find the slope of the tangent line to the curve f(x)=x2+3x+2f(x) = x^2 + 3x + 2f(x)=x2+3x+2 at $x=1$.

1. Find the derivative:

   f′(x)=ddx(x2+3x+2)=2x+3f'(x) = \frac{d}{dx}(x^2 + 3x + 2) = 2x + 3f′(x)=dxd​(x2+3x+2)=2x+3

2. Evaluate at $x=1$:

   f′(1)=2(1)+3=5f'(1) = 2(1) + 3 = 5f′(1)=2(1)+3=5

Result: The slope of the tangent line at $x=1$ is 5.

Example 2: Trigonometric Function

Find the slope of the tangent line to $f(x)=\sin (x)$ at x=π4x = \frac{\pi}{4}x=4π​.

1. Find the Derivative:

   f′(x)=cos⁡(x)f'(x) = \cos(x)f′(x)=cos(x)

2. Evaluate at x=π4x = \frac{\pi}{4}x=4π​:

   f′(π4)=cos⁡(π4)=22f’\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}f′(4π​)=cos(4π​)=22​​

Result: The slope of the tangent line at x=π4x = \frac{\pi}{4}x=4π​ is 22\frac{\sqrt{2}}{2}22​​.

Key Points

1. Geometric Meaning: The tangent line touches the curve at exactly one point and has the same slope as the curve at that point.
2. Derivative as Rate of Change: The derivative represents how the function changes at a particular point, which is why it gives the slope of the tangent line.
3. Applications: Tangent line slopes are used in physics, optimization, and curve analysis.

By following these steps, you can find the slope of the tangent line for any differentiable function.