How to Find the Slope of a Tangent Line?

To find the slope of the tangent line, we should have a clear concept of tangent lines and slope. The slope is defined as the ratio of the difference in y coordinate to the difference in x coordinate. It is represented by the following formula:

m =( y⁠⁠⁠⁠⁠⁠⁠⁠⁠⁠⁠⁠⁠⁠₂ – y⁠⁠⁠⁠⁠⁠⁠₁ ) /(x⁠⁠⁠⁠⁠⁠⁠₂  – x⁠⁠⁠⁠⁠⁠⁠₁)

It is to be noted that: 

• tan θ is the same as m. Slopes can be positive or negative depending on whether the line is moving up or down. 
• Products of the slope of two perpendicular lines are -1 and slopes in parallel lines are the same. 
• Derivative of a function gives a change in rate with respect to change in the independent variable. 

Slope of a Tangent Line

The tangent line is the line that touches a curve at a point. There may be tangent lines that later cross the curve or touch the curve at some other points.

But the basic criteria for a line to be a tangent line of curve f(x) at a point x=a if the line passes through the point (a, f(a)) (where the point is common both to the curve and the tangent line) and the tangent line has slope f'(a) where f'(a) is derivative of function f(x) at point a. 

The slope of the tangent line is the same as the derivative of the curve at some point. The formula for a tangent line whose slope is m and the point given is (x⁠⁠⁠⁠⁠⁠⁠₁, y⁠⁠⁠⁠⁠⁠⁠₁ ) is given by,

y – y⁠⁠⁠⁠⁠⁠⁠₁ = m × (x – x⁠⁠⁠⁠⁠⁠⁠₁ ) 

or 

y= mx + c 

Where c is some constant.

Read More about Slope of a Line.

How to Find the Slope of a Tangent Line?

Solution:

The slope of a tangent line can be found by finding the derivative of the curve f(x and finding the value of the derivative at the point where the tangent line and the curve meet. This gives us the slope

For example: Find the slope of the tangent line to the curve f(x) = x² at the point(1, 2). Also, find the equation of the tangent line. 

Let us find derivative of f(x):

f'(x) = dy/dx = d(x²) /dx = 2x

Value of slope at point(1, 2) is,

f'(x) = 2(1) = 2

The equation of tangent line is 

y – 2 = 2(x – 1) 

or 

y = 2x

Read Also,

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

Similar Problems

Problem 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

Therefore the value of the slope is 0.5.

Problem 2: Find the slope given two points (6, 7) and (8, 0).

Solution: 

Slope of any two points say (a, b) and (x, y) is given by,

m = (y-b) /(x-a) 

Therefore m = (0-7) /(8-6) = -3.5

Problem 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

Problem 4: Find the slope of 2 lines that are perpendicular to each other given 1 equation is y= 3x+8

Solution: 

Let the slope of two perpendicular lines be m and n

m×n = -1

⇒ m = 3

⇒ n = -1/3

Problem 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)