The slope between two points on a curve is a secant slope — an average rate of change.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Bring them together
Slide the second point towards the first. The secant slopes approach a single number.
That number is the derivative
f'(a) is the limit of (f(a + h) − f(a)) / h as h approaches 0.
What it means
It is the slope of the tangent at that point, and the instantaneous rate of change there.
The definition
f′(a) is the limit as h approaches zero of (f(a + h) − f(a))/h. The exam asks for this form explicitly, and being able to recognise a given limit as a derivative in disguise is a standard question type.
The alternative form
f′(a) is also the limit as x approaches a of (f(x) − f(a))/(x − a). The two forms are the same limit written with different variables, and either may appear on the exam.
Three readings of the same number
Slope of the tangent line, instantaneous rate of change, and velocity when the function is position. Free-response questions ask for the interpretation in context, with units, not just the value.
Units are part of the answer
A derivative's units are output units per input unit. If the answer is a rate of water flow, "12 litres per minute" is the answer and "12" is not. Omitting units costs marks routinely.
Step 2: Try It Yourself
Drag the second point onto the first and watch the secant become the tangent.
- Point(2, 4)
- Second point(5, 25)
- Slope between them7
Slide the second point towards the first. The slope between them approaches the slope of the curve at that point.
Step 3: Watch an Example
One step at a time.
Watch Nadia Take the Limit
Nadia computes f'(3) for f(x) = x² from the definition.
- Step 1
The difference quotient is ((3 + h)² − 9) / h.
Step 4: Your Turn
Practice makes it stick.
From the Definition
Problem 1 of 2
For f(x) = x², what is f'(5)?
Reading the Tangent
Problem 2 of 2
The tangent to y = x² at x = 2 is shown. What is its slope?
- Point(2, 4)
- Slope of the tangent4
a
b
c
Point of tangency
Secant to Tangent
1 of 8
f(x) = x². What is f'(1)?
2 of 8
f(x) = x². What is f'(0)?
3 of 8
f(x) = 5x + 2. What is f'(9)?
4 of 8
Set the two points together at x = 2 so the secant becomes the tangent, and the slope reads 4.
- Point(-3, 9)
- Second point(5, 25)
- Slope between them2
Slide the second point towards the first. The slope between them approaches the slope of the curve at that point.
a
b
c
5 of 8
The difference quotient for f(x) = x² at a = 4 simplifies to 8 + h. What is the derivative?
6 of 8
f(x) = x³. What is f'(2)? The derivative of x³ is 3x².
7 of 8
Is the derivative an average rate or an instantaneous rate? 1 for average, 2 for instantaneous.
8 of 8
f(x) = 7. What is f'(3)?
Step 5: Quick Check
Show what you know.
Question 1 of 1
f(x) = x². What is f'(7)?
What You Learned
- A secant slope is an average rate of change between two points.
- The derivative is the limit of that slope as the points come together.
- It is the tangent slope, and the instantaneous rate of change.