Finding the Equation of a Tangent Line with Derivatives

Most students, particularly high school students, find finding the equation of a tangent line with derivatives challenging.

You might wonder what tangent lines are exactly. In simple terms, a tangent line is a line that touches a curve at exactly one point.

Read on to explore the process of finding the equation of a tangent line using derivatives step by step. You might also enjoy reading: Gradients and Areas Under Graphs, what they mean, and why are they important.

What a Tangent Line Is?

In calculus, a tangent line is a straight line that touches a curve at just one point. Finding the tangent line at a point on a curve can tell you about the slope of that curve at that point.

In other words, the slope of the tangent line shows how steep the curve is at that specific point. If you are interested in exploring more tangent lines, I wrote a whole article where I discussed what a tangent line is and everything you need to know about finding a tangent line.

Step 1: Find The Point Where The Tangent Line Intersects The Main Function

To start, we need to find the point where the tangent line intersects the main function, which is also called the point of tangency or, more commonly, the point of contact.

In other words, we need to find the value of x where the tangent line intersects the curve.

Step 2: Take The Derivative of The Function to Find Its Slope at The Point of Tangency

Now, you need to find the derivative of the function to find its slope at the point of tangency, which will give you the slope of the tangent line at the exact point we’re looking for.

You can take the derivative using various methods, but the easiest way is by using the power rule. The power rule states that if you have f(x) = xn, then f'(x) = nx^n-1.

Step 3: Use The Equation of a Tangent Line Formula y = f'(a)(x – a) + f(a)

After finding the slope of the tangent line, we can either use the following:

1- The equation of a tangent line formula y = f'(a)(x – a) + f(a)

Example: Suppose our function is y = x², and we want to find the equation of the tangent line at the point x= 2, meaning that a = 2.

• f'(x) = 2x
• f'(2) = 4
• f(2) = 4

Using the equation of a tangent line formula y = f'(a)(x – a) + f(a), we will find:

• y = 4(x-2) + 4, and y = 4x – 4

2- Using the point-slope formula to write the equation of the tangent line

We know that the equation of a line is y = mx + b, where m is the slope and b is the y-intercept. However, for our tangent line, we need to use the point-slope formula, which is y – y₁ = m(x – x₁), where (x₁, y₁) is the point of tangency.

Example: Suppose our function is y = x², and we want to find the equation of the tangent line at the point x= 2

• dy/dx= 2x
• f'(2) = 4

Using the point-slope formula to write the equation of the tangent line, we will find y – 4 = 4(x – 2), which simplifies to y = 4x – 4.

To learn more, I encourage you to watch this video or the video below.

What to read next:

• Solving Systems of Equations by Substitution Method.
• Understanding the FOIL Method for Binomial Multiplication.
• How To Factor Polynomials (including 5 methods with examples)

Final Thoughts

Finding the equation of a tangent line with derivatives may seem challenging at first glance, but with practice and understanding the principles, you will find it easy.

By following the steps I have outlined, you will ace your next exam on finding the equation of a tangent line, which you will find easy to calculate the equation of the tangent line at any point on the curve.