The slope of a line needs two points. A tangent touches at one, so the usual formula has nothing to work with.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Start with a secant
Take a second point nearby and find the slope of the line through both. That is a secant slope, and it is an approximation.
Then close the gap
Slide the second point toward the first. The secant slopes settle on a single value, which is the tangent slope.
The difference quotient
That process is the limit of [f(x + h) − f(x)] ÷ h as h approaches zero. This expression is the derivative.
What it measures
The tangent slope is the instantaneous rate of change: the speed at one moment rather than the average over an interval.
Why the limit is essential
Setting h = 0 outright gives 0 ÷ 0. Only the limit gets past that, which is why this chapter came first.
Where calculus begins
The slope between two points on a curve is a straightforward calculation. The slope at a single point is not — there is only one point, so there is no rise and no run. Resolving that is the tangent line problem.
Secants approaching a tangent
Take a second point nearby and compute the secant slope. Move it closer and the secants approach the tangent. The tangent slope is the limit of the secant slopes, which is precisely a limit calculation.
The difference quotient
(f(a + h) − f(a))/h is the secant slope over an interval of width h. Taking the limit as h approaches zero gives the derivative. Everything in differential calculus is built on this one expression.
Why the limit is unavoidable
Setting h to zero directly gives 0/0, which says nothing. Only the limiting process extracts a definite answer. This is the concrete problem that forced limits to be made rigorous in the first place.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 4)
- Slope of the tangent4
Step 3: Watch an Example
One step at a time.
Watch Yusuf Close In on a Tangent Slope
Yusuf wants the tangent slope of y = x² at x = 2.
- Step 1
He takes a second point at x = 3, giving a secant slope of (9 − 4) ÷ 1 = 5.
Step 4: Your Turn
Practice makes it stick.
The Secant
Problem 1 of 2
On y = x², the points at x = 1 and x = 3. What is the secant slope?
The Tangent
Problem 2 of 2
On y = x², the tangent slope is 2x. What is it at x = 5?
Close the Gap
1 of 8
On y = x², tangent slope at x = 3?
2 of 8
On y = x², tangent slope at x = 0?
3 of 8
On y = x², tangent slope at x = −4?
4 of 8
On y = x², points at x = 2 and x = 4. Secant slope?
5 of 8
On the line y = 3x + 1, what is the tangent slope anywhere?
6 of 8
Setting h = 0 in the difference quotient gives which form? Enter 1 for 0 ÷ 0, 2 for a number.
7 of 8
Put the tangent line process in order.
- 1Write the secant slope as a difference quotient.
- 2Simplify until h cancels from the denominator.
- 3Take the limit as h approaches zero.
- 4Choose a second point a distance h away.
8 of 8
On y = x², tangent slope at x = 7?
Step 5: Quick Check
Show what you know.
Question 1 of 2
On y = x², what is the tangent slope at x = 6?
Question 2 of 2
Why must a limit be used rather than simply setting h = 0?
What You Learned
- A tangent slope is the limit of secant slopes as the second point closes in.
- That limit of [f(x + h) − f(x)] ÷ h is the derivative.
- It measures the instantaneous rate of change.