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Understanding the different ways to write derivatives
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There are several ways to write derivatives. They all mean the same thing, but are used in different contexts!
For a function , all of these mean "the derivative":
| Notation | Read as | Context |
|---|---|---|
If , write the derivative using three different notations.
| Section | Format | Questions | Time | Weight | Calculator |
|---|---|---|---|---|---|
| Multiple Choice (No Calculator) | MCQ | 30 | 60 min | 33.3% | 🚫 |
| Multiple Choice (Calculator) | MCQ | 15 | 45 min | 16.7% | ✅ |
| Free Response (Calculator) | FRQ | 2 | 30 min | 16.7% | ✅ |
| Free Response (No Calculator) | FRQ | 4 | 60 min | 33.3% | 🚫 |
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A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of cm/s. How fast is the area of the circle increasing when the radius is cm?
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| "f prime of x" |
| Lagrange notation |
| "dy dx" | Leibniz notation |
| "y prime" | When y is the function |
| "df dx" | Alternative Leibniz |
| "D f of x" | Operator notation |
| "y dot" | Physics (time derivative) |
Prime notation is the most common in calculus.
Example: If , then
This looks like a fraction, and sometimes we can treat it like one!
Parts:
Example: If , then
To indicate the derivative at a particular value:
Lagrange:
Leibniz:
The vertical bar means "evaluated at".
Second derivative (derivative of the derivative):
| Notation | Meaning |
|---|---|
| Second derivative | |
| Second derivative (Leibniz) | |
| Second derivative |
Third derivative:
nth derivative:
Use when:
Use when:
Use when:
For :
Prime notation:
Leibniz notation:
At :
They all give the same answer!
When we write , we can think of it as an operator:
Read as: "Take the derivative with respect to x of "
For functions of multiple variables (later in calculus):
The symbol indicates a partial derivative (holding other variables constant).
First, find the derivative: The derivative of is , and the derivative of is .
Three notations:
Lagrange (prime) notation:
Leibniz notation (if we call the function y):
All three express the same derivative!
What is the difference between f''(x) and d²y/dx²?
There is no difference in meaning!
Both represent the second derivative - the derivative of the derivative.
f''(x) is Lagrange notation (prime notation):
d²y/dx² is Leibniz notation:
Example: If f(x) = x⁴:
If y = x³, write the derivative in four different notations.
Step 1: Find the derivative: dy/dx = 3x²
Step 2: Leibniz notation: dy/dx = 3x²
Step 3: Lagrange (prime) notation: y' = 3x² or f'(x) = 3x²
Step 4: Newton (dot) notation: ẏ = 3x² (used mainly in physics)
Step 5: Operator notation: Dx[x³] = 3x² or d/dx[x³] = 3x²
Answer: dy/dx = y' = f'(x) = ẏ = 3x²
If f(x) = 2x² - 5x + 1, find f'(3) and interpret what it means.
Step 1: Find f'(x): f'(x) = 4x - 5
Step 2: Evaluate at x = 3: f'(3) = 4(3) - 5 = 12 - 5 = 7
Step 3: Interpret: At x = 3, the slope of the tangent line is 7
Step 4: Rate of change interpretation: At x = 3, f(x) is increasing at a rate of 7 units vertically per 1 unit horizontally
Step 5: Alternative notations: dy/dx|ₓ₌₃ = 7 or Df(3) = 7
Answer: f'(3) = 7 (slope of tangent at x = 3)
If s(t) = 16t² represents position in feet, what does s'(2) = 64 mean in context?
Step 1: Understand the notation: s(t) = position function s'(t) = velocity function (rate of change of position) t = time
Step 2: Given information: s'(2) = 64
Step 3: Interpretation: At t = 2 seconds, the velocity is 64 feet per second
Step 4: Alternative notations meaning the same thing: ds/dt|ₜ₌₂ = 64 v(2) = 64 (if v represents velocity)
Step 5: Physical meaning: At exactly 2 seconds, the object is moving at an instantaneous rate of 64 ft/s
Answer: At t = 2 seconds, the velocity is 64 feet per second
Operator notation: