How to Find the Slope of a Tangent Line?

To find the slope of a tangent line to a curve at a given point, you need to find the derivative at that point. The derivative represents the instantaneous rate of change of the function — in other words, it's the slope of the tangent line.

To find the slope of the tangent line to a function f(x) at a point x = a, use this formula:

Slope of tangent line = f^'(a)

To find the slope of a tangent line to a curve at a given point, follow these steps:

Step-by-Step Process

1. Understand the Tangent Line: A tangent line touches a curve at a single point and has the same slope as the curve at that point.
2. Find the Derivative: The derivative of the function f (x), denoted as f′(x), represents the slope of the curve at any point x.
3. Evaluate the Derivative at the Point: Substitute the x-coordinate of the given point into f′(x) to find the slope of the tangent line at that specific point.

Example: Find the slope of the tangent line to f(x) = x² at (1, 1):
Solution:

1. Derivative: f′(x) = 2
2. Evaluate at x = 1: f′(1) = 2(1) = 2
3. Slope: The slope of the tangent line is 2.

Slope of a Tangent Line

The slope of a tangent line tells us how steep the curve is at a specific point. Imagine standing on the curve at that point — the slope of the tangent line is like the slope of the ground under your feet. It shows the instantaneous rate of change of the function at that point.

The slope of the tangent line is essentially the rate of change of the function, which is given by the derivative. The derivative of a function f(x) at a point x = a is the slope of the tangent line to the curve at that point.

Understanding the Tangent Line

A tangent line touches a curve at exactly one point and has the same slope as the curve at that point. To find the slope of this line, we use the concept of limits and derivatives.

Formula for the Slope of the Tangent Line

The slope of the tangent line to a curve y = f(x)at a point (x₀, y₀) is given by:

m = f^' (x₀)

wheref^'(x) is the derivative of f(x), and x₀ is the x-coordinate of the point.

Steps to Find the Slope

1. Find the Derivative: Compute f^'(x), which represents the slope of the curve at any point x.
2. Substitute the Point: Plug x₀ f^'(x) to calculate m =f^'(x₀), the slope at that specific point.
3. Tangent Line Equation: Use the point-slope form: y-y₀ = m (x-x₀)

Example: Find the slope of the tangent line to f(x) = x² + 3x + 2 at x = 1.

Solution:

1. Derivative: f^'(x) = 2x+3
2. Substitute x = 1: f′(1) = 2(1) + 3 = 5
3. Slope: The slope of the tangent line is m = 5.
4. Tangent Line Equation: At (1, f (1)) = (1, 6):
   y−6 = 5(x - 1)
   y = 5x + 1

Read more about: Slope of a Line.

For example: Find the slope of the tangent line to the curve f(x) = x² at the point(1, 1). Also, find the equation of the tangent line.

Solution:

To find the slope of the tangent line, we need to find f'(x):

f(x) = x²
f'(x) = dy/dx = d(x²) /dx = 2x

At x = 1, the slope will be: f'(1) = 2(1) = 2

So, the slope of the tangent line is 2.

Now for the equation of the tangent line: We'll use point-slope form: y - y₁ = m(x - x₁)

We have point (1, 1) and slope = 2
y - 1 = 2(x - 1)

Convert to slope-intercept form:
y - 1 = 2x - 2
y = 2x - 1

• The slope of the tangent line is 2
• The equation of the tangent line is y = 2x - 1

Read Also,

• Tangents and Normals
• Slope of the Secant Line Formula
• How to Find Slope From a Graph?

Solved Question on Slope of a Tangent Line

Question 1: Find the slope of the tangent line 6y = 3x + 5.
Solution: 

Since we know the equation of a tangent line is of the form y= mx + c where m is the slope

We can write, 
y= (3x + 5 ) / 6
y= (3/6(x ) + 5 /6)
y= (1/2(x ) + 5 /6)

Therefore the value of the slope is 1/2 = 0.5.

Question 2: Find the slope of the tangent line to the curve y = 3x² − 4x + 1 at x=2.
Solution: 

Find f^'(x) : f′(x) = d/dx (3x²− 4x + 1) = 6x−4
Substitute x=2: f′(2) = 6(2)− 4 = 8
Slope:
The slope of the tangent line at x=2 is 8.

Question 3: Find the slope of the curve y = 6x³.
Solution:

The slope of curve is given by differentiation of the curve:

dy/dx = d(6x³) /dx = 18x²

Question 4: Find the slope of the tangent line to the curve f(x) = x² + 2x at x = 3.
Solution: 

Find f^'(x) : f′(x) = d/dx (x²+ 2x) = 2x + 2
Substitute x=3: f′(3) = 2(3) + 2 = 8
Slope:
The slope of the tangent line at x=3 is 8.

Question 5: Find the slope of the tangent line to the curve f(x) = x⁴ at the point(2, 1). Also, find the equation of the tangent line. 
Solution:

Let us find the derivative of the curve as,

dy/dx = 4x³

At point (2, 1) value of dy/dx or slope m is,

m = 32

Equation of tangent line at point (2, 1) is,
y - 1 = 32(x - 2) 
y - 1 = 32x - 64
y = 32x - 63

Unsolved Question on Find the Slope of a Tangent Line

Question 1: Find the slope of the tangent line √2/2√√e to the curve f(x) = 2x² + x at x = 3.

Question 2: Find the slope of the tangent line to f(x) = 1/x at x = 2.

Question 3: Find the slope of the tangent line to the curve f(x) = sin(x) at x = π/4.

Question 4: Find the slope of the tangent line to the curve f(x) = ln(x) at x = 1.

Answer Key

1. 13
2. -1/4
3. √2/2
4. 1