Differentiation from First Principles
Understand how the derivative is defined using limits, starting from the gradient of a secant line and taking the limit as the two points converge.
From Secant to Tangent
The gradient of a secant line through two points on a curve gives an average rate of change. As the two points get closer together, the secant line approaches the tangent line, and its gradient approaches the instantaneous rate of change.
Visual: Secant Approaching Tangent
Large h: secant line
Smaller h: closer
h → 0: tangent line
Gradient of secant through (x, f(x)) and (x + h, f(x + h)):
msecant = f(x + h) − f(x)h
The Formal Definition of the Derivative
The derivative of f(x) is defined as the limit of the secant gradient as h approaches 0:
f′(x) = limh→0 f(x + h) − f(x)h
This is the "first principles" definition of the derivative
The process for finding a derivative from first principles is:
Write out f(x + h) by substituting (x + h) for every x in the function.
Calculate f(x + h) − f(x) and simplify.
Divide by h and cancel the common factor.
Take the limit as h → 0.
Derivative Notation
There are several equivalent ways to write the derivative. You will encounter all of these in the HSC:
| Notation | Read as | Named after |
|---|---|---|
| f′(x) | "f prime of x" or "f dash of x" | Lagrange notation |
| dydx | "dy by dx" | Leibniz notation |
| y′ | "y prime" or "y dash" | Newton notation |
Key Vocabulary
Derivative
The instantaneous rate of change of a function, equal to the gradient of the tangent line.
First Principles
The method of finding the derivative using the limit definition directly.
Secant Line
A straight line that passes through two points on a curve.
Tangent Line
A straight line that touches the curve at exactly one point and has the same gradient as the curve there.
Worked Examples
Find f′(x) from first principles if f(x) = x².
Step 1: f(x + h) = (x + h)² = x² + 2xh + h².
Step 2: f(x + h) − f(x) = x² + 2xh + h² − x² = 2xh + h².
Step 3: f(x+h) − f(x)/h = 2xh + h²/h = 2x + h.
Step 4: limh→0 (2x + h) = 2x. So f′(x) = 2x.
Find f′(x) from first principles if f(x) = 3x + 5.
Step 1: f(x + h) = 3(x + h) + 5 = 3x + 3h + 5.
Step 2: f(x + h) − f(x) = (3x + 3h + 5) − (3x + 5) = 3h.
Step 3: 3h/h = 3.
Step 4: limh→0 3 = 3. The gradient is constant for a linear function.
Find f′(x) from first principles if f(x) = x³.
Step 1: f(x + h) = (x + h)³ = x³ + 3x²h + 3xh² + h³.
Step 2: f(x + h) − f(x) = 3x²h + 3xh² + h³.
Step 3: Divide by h: 3x² + 3xh + h².
Step 4: limh→0 (3x² + 3xh + h²) = 3x².
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the first principles definition of the derivative?
Question 2
Using first principles, if f(x) = 5x, what is f′(x)?
Question 3
Using first principles, what is the derivative of f(x) = x² + 3x?
Question 4
What does the derivative f′(a) represent geometrically?
Question 5
If f(x) = 7 (a constant), what is f′(x) from first principles?
Key Concepts Summary
- ●The derivative is defined as f′(x) = limh→0 [f(x+h) − f(x)] / h.
- ●Geometrically, the derivative gives the gradient of the tangent line.
- ●Steps: substitute (x+h), expand, subtract f(x), divide by h, take the limit.
- ●The derivative of a constant is 0 and the derivative of xn is nxn−1.
- ●Common notations: f′(x), dy/dx, and y′.