high school 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 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

④Differentiable implies continuous; the converse fails. |x| is continuous at x=0 but not differentiable

From Secant to Tangent

Position a1

Δx (gap)1.5

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

(xⁿ)' = nx^n-1

Math II uses natural-number n. Real exponents are Calculus. 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

(fg)' = f'g - fg'g²

(top′ × bottom − top × bottom′) / bottom². This rule is on Calculus

Meaning of f'(x)

x position0

💡 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).

Substitute into the definition: f'(2) = lim_h→0 [f(2+h) − f(2)]/h.

f'(2) = lim_h→0 (2+h)² - 2²h

Expand, cancel, then let h → 0.

= lim_h→0 4h + h²h = 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.

Apply the power rule (xⁿ)' = nx^n-1 to each term.

(x³)' = 3x², (2x²)' = 4x, (5)' = 0

Combine term by term.

f'(x) = 3x² - 4x

▸ f'(x) = 3x² − 4x

Differentiate a polynomial term by term with the power rule (the derivative of a constant is 0).

exam-style

For f(x) = x² + 3x, what is the derivative f'(1) at x = 1?

①5

②4

③2

④3

⑤8

▸ ① 5

Find the derivative: f'(x) = 2x + 3.

f'(x) = (x²)' + (3x)' = 2x + 3

Substitute x = 1.

f'(1) = 2(1) + 3 = 5

← Previous

Continuity of Functions

Applications of Derivatives

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