In calculus, we learn that the tangent line for a function can be found by computing the derivative. So there’s a close relationship between derivatives and tangent lines. However, they are not the same thing. For starters, the derivative f ‘(x) is a function, while the tangent line is, well, a line.

Calculus Applets with GeoGebra. 2 Riemann Sums and Integrals Tangent Line as a Limit of Secant Lines Trace Derivative About. Line is a tangent to the function. Question 2 From rst principles, nd the derivative f0(t) of the function f(t) = p t. Hence nd the point on the graph y = p t, where the tangent line has slope 1 4. Also nd the equation of that tangent line and sketch the tangent line. Find the derivative of the function. F(x) = (2x – 5)^4(x^2 + x + 1)^5 6. Find the derivative of the function. Y = ((x^2 + 5)/(x^2 – 5))^4 7. Find the derivative of the function. Y = sin(tan 4x) 8. Find the derivative of the function. Y = x sin( 3/x) 9. Find the derivative of the function. Y= cot^2(cos theta) 10. The derivative (dy/dx) will give you the gradient (slope) of the curve. Find a value of x that makes dy/dx infinite; you’re looking for an infinite slope, so the vertical tangent of the curve is a vertical line at this value of x. Vertical Tangent in Calculus Example. Example Problem: Find the vertical tangent of the curve y = √(x – 2). Very frequently in beginning Calculus you will be asked to find an equation for the line tangent to a curve at a particular point. We’re calling that point $(x0, y0)$. To find the line’s equation, you just need to remember that the tangent line to the curve has slope equal to the derivative of the function evaluated at the point of interest.

Instead, the correct statement is this: “The derivative measures the slope of the tangent lines.”

Think about this: a clock is not the same thing as time. But if you want to know the time of day, you can go look at a clock to find out. A clock measures the time at any particular point throughout the day.

Let’s take a closer look at tangent lines.

What is a tangent Line?

A tangent line for a function f(x) at a given point x = a is a line (linear function) that meets the graph of the function at x = a and has the same slope as the curve does at that point.

2.5 Derivative As Tangent Lineup Calculus Equation

Sometimes we might say that a tangent line “just touches” the curve, or “intersects the curve only once,”f but those ideas can sometimes lead us astray.

The graph below shows the tangent lines (in red, purple, and magenta) at three different points on a curve y = f(x) (in black).

Slope and Derivatives

So how do we know what the slope of the tangent line should be? After learning about derivatives, you get to use the simple formula,

m = f ‘(a).

In this formula, the function f and x-value a are given. Your job is to find m, which represents the slope of the tangent line. Once you have the slope, writing the equation of the tangent line is fairly straightforward.

Finding the Tangent Line

Suppose you are asked to find the tangent line for a function f(x) at a given point x = a. Here is a step-by-step approach:

1. Find the derivative, f ‘(x).
2. Plug in x = a to get the slope. That is, compute m = f ‘(a).
3. If not already given in the problem, find the y-coordinate of the point. As always, you plug the x-value into the function in order to get the y-value. Let b = f(a).
4. Use the point-slope form and solve for y to find the equation of the tangent line. In other words, plug in your values of m, a, and b into the equation,

   y = m(x – a) + b.

The Derivative Measures Slope

Let’s take another look at that first step, “Find the derivative.” Remember, the derivative is a function (of the input variable x). By plugging in different input values, x = a, the output values of f ‘(x) give you the slopes of the tangent lines at each point x = a.

This is what we mean when we say that “the derivative measures the slope of the tangent lines.”

If I want to know the slope of f at x = 1, then I compute f ‘(1). And if I want to know the slope at x = -352/13, then I compute f ‘(-352/13). Simple as that!

Example: A Polynomial

Now let’s look at an example function, f(x) = x³ + 3x² + 1. We’ll find the tangent lines at a few different points.

First of all, find the derivative: f ‘(x) = 3x² + 6x. (for practice finding derivatives, check out this Magoosh article about derivatives).

The function value and derivative value at a few points are shown in the table below:

2.5 Derivative As Tangent Lineup Calculus Calculator

For the points listed, we can easily find the equation of the tangent line. Note, on the AP Calculus Exam, the multiple choice answers may be simplified. So I’ve shown you both the point-slope form and the simplified, or slope-intercept, form of the tangent line.

Now let’s see the graph of y = f(x) together with the tangent lines that we just found.

2.5 Derivative As Tangent Lineup Calculus Solver

Summary

So now you know. The derivative is not the same thing as a tangent line. Instead, the derivative is a tool for measuring the slope of the tangent line at any particular point, just like a clock measures times throughout the day. With this in mind, you’ll have no trouble tackling tangent line problems on the AP Calculus exam!

For more about slope, tangent lines, and derivatives, check out these related Magoosh articles: Is the Derivative of a Function the Slope? and How to Find the Slope of a Line Tangent to a Curve.

About Shaun Ault

Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!

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2.5 Derivative As Tangent Lineup Calculus Formula

Motivating Questions

• What is the formula for the general tangent line approximation to a differentiable function (y = f(x)) at the point ((a,f(a))text{?})

• What is the principle of local linearity and what is the local linearization of a differentiable function (f) at a point ((a,f(a))text{?})

• How does knowing just the tangent line approximation tell us information about the behavior of the original function itself near the point of approximation? How does knowing the second derivative's value at this point provide us additional knowledge of the original function's behavior?