Derivative and Differentiation Derivative and Differentiation
Differentiation is the tool that pins down exactly how fast a function is changing at this very instant.
You start from the slope of a secant line through two points (the average rate) and shrink the gap between them toward zero, until it becomes the slope of a tangent line that just grazes the curve (the instantaneous rate).
That tangent slope is precisely the derivative at that point.
Shrink the Δx slider and watch the exact moment the blue secant slides right onto the orange tangent.
Core Idea of Differentiation
💡 What is Differentiation?
①Computing the "instantaneous rate of change"
②Average rate (secant slope) → as Δx → 0 → instantaneous rate (tangent slope)
③The exact slope at a point on a curve
From Secant to Tangent
see gongsik.com
As Δx → 0, the blue secant converges to the orange tangent Definition of Derivative
f'(a) = lim Δx→0 f(a + Δx) - f(a) Δx
Instantaneous rate at x = a = slope of the tangent line
🔑 Secant → Tangent
①Secant: line through two points → average rate
②As Δx shrinks, the second point approaches the first
③Δx → 0 limit: secant becomes tangent → instantaneous rate!
Basic Differentiation Rules
Power Rule
(x n )' = nx n-1
Holds for real n — the foundation of differentiation
Constant Multiple / Sum / Difference
(cf)' = cf', (f ± g)' = f' ± g'
Constants pull out; sums and differences split
Product Rule
(fg)' = f'g + fg'
Differentiate one, leave the other; add them up
Quotient Rule
( f g )' = f'g - fg' g²
(top′ × bottom − top × bottom′) / bottom²
Meaning of f'(x)
see gongsik.com
Top: f(x); bottom: f'(x) — observe how slope changes with x 💡 Relationship between f(x) and f'(x)
①f'(x) > 0 → f(x) increasing
②f'(x) < 0 → f(x) decreasing
③f'(x) = 0 → candidate for local max/min
Wrap-up
Derivative
f'(a) = lim h→0 f(a+h) - f(a) h
Slope of the tangent line at x = a
🎯 Exam Points
①Differentiable ⇒ continuous (converse fails: |x| at x=0)
②(x^n)' = nx^{n-1} is the foundation
③Product rule: f'g + fg'
④Critical point f'(a) = 0 — check sign change
⑤Tangent line: y − f(a) = f'(a)(x − a)
Worked Examples & Past Exam
Example 1
For f(x) = x², use the definition of the derivative to find f'(2).
1 Substitute into the definition: f'(2) = lim h→0 [f(2+h) − f(2)]/h.
f'(2) = lim h→0 (2+h) 2 - 2 2 h
2 Expand, cancel, then let h → 0.
= lim h→0 4h + h 2 h = lim h→0 (4 + h) = 4
▸ f'(2) = 4
The derivative is the slope of the tangent; the definition is the limit of the average rate.
Example 2
Find the derivative f'(x) of f(x) = x³ − 2x² + 5.
1 Apply the power rule (x n )' = nx n-1 to each term.
(x 3 )' = 3x 2 , (2x 2 )' = 4x, (5)' = 0
2 Combine term by term.
f'(x) = 3x 2 - 4x
▸ f'(x) = 3x² − 4x
Differentiate a polynomial term by term with the power rule (the derivative of a constant is 0).
2023 CSAT Math type, adapted
For f(x) = x² + 3x, what is the derivative f'(1) at x = 1?
▸ ① 5
1 Find the derivative: f'(x) = 2x + 3.
f'(x) = (x 2 )' + (3x)' = 2x + 3
2 Substitute x = 1.
f'(1) = 2(1) + 3 = 5