Calculus — The Definition of the Derivative and Differentiation Rules

The Definition of the Derivative

The limit definition of the derivative:
f'(x) = lim [f(x+h) − f(x)] / h
         h→0

Intuitive meaning:
→ f'(x): the instantaneous rate of change of f at the point x
→ The slope of the tangent line at (x, f(x)) on the curve y = f(x)

Notation:
→ f'(x) (Lagrange notation)
→ df/dx (Leibniz notation)
→ Df(x) (Euler notation)
→ ẋ (Newton notation, for differentiation with respect to time)

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The Geometric Meaning of the Derivative

Secant line:

• The line through the two points (x, f(x)) and (x+h, f(x+h))
• Slope = [f(x+h) − f(x)] / h (the average rate of change)

Tangent line:

• As h → 0, the secant line converges to the tangent line
• The tangent’s slope = f’(x) (the instantaneous rate of change)

Physical meaning:

• For a position function s(t): velocity = s’(t) = ds/dt
• For a velocity function v(t): acceleration = v’(t) = a(t)

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Differentiability and Continuity

Condition for differentiability:

• lim [f(a+h) − f(a)] / h exists at x = a h→0

The relationship between continuity and differentiability:

• If f is differentiable at x=a → f is continuous at x=a (the converse does not hold)

An example that is continuous but not differentiable:

• f(x) = |x| (the absolute-value function)
• Continuous at x=0, but not differentiable there
• The slope of the left-hand tangent = −1
• The slope of the right-hand tangent = +1
• They differ, so the function is not differentiable

Cases where a function is not differentiable:

1. A corner: shapes like |x|
2. A cusp: shapes like x^(2/3)
3. A vertical tangent: f’(x) = ±∞
4. A point of discontinuity

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Basic Differentiation Formulas

Constant: (c)' = 0
Power rule: (xⁿ)' = nxⁿ⁻¹
Constant multiple: (cf)' = cf'
Sum/difference: (f ± g)' = f' ± g'

Examples:
→ (x³)' = 3x²
→ (5x²)' = 10x
→ (x³ + 2x)' = 3x² + 2
→ (√x)' = (x^(1/2))' = (1/2)x^(−1/2) = 1/(2√x)

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The Product Rule and the Quotient Rule

Product rule:

(fg)’ = f’g + fg’

Example:

• y = x² · sin(x)
• y’ = 2x · sin(x) + x² · cos(x)

Quotient rule:

(f/g)’ = (f’g − fg’) / g²

A mnemonic: “low d-high minus high d-low, over the square of what’s below”

Example:

• y = sin(x) / x
• y’ = (cos(x)·x − sin(x)·1) / x² = (x·cos(x) − sin(x)) / x²

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The Chain Rule

The chain rule:

d/dx [f(g(x))] = f’(g(x)) · g’(x)

In Leibniz notation:

dy/dx = (dy/du) · (du/dx)

Example 1: y = sin(x²)

• f(u) = sin(u), g(x) = x²
• y’ = cos(u) · 2x = cos(x²) · 2x = 2x·cos(x²)

Example 2: y = (3x + 1)⁵

• f(u) = u⁵, g(x) = 3x + 1
• y’ = 5u⁴ · 3 = 5(3x+1)⁴ · 3 = 15(3x+1)⁴

Example 3: y = √(x² + 1)

• y = (x² + 1)^(1/2)
• y’ = (1/2)(x² + 1)^(−1/2) · 2x = x / √(x² + 1)

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Differentiating Trigonometric Functions

$$\begin{aligned} &\text{Basic trigonometric derivatives}: \\ (\text{sin} x)' &= \text{cos} x \\ (\text{cos} x)' &= −\text{sin} x \\ (\text{tan} x)' &= \text{sec}^2x = \frac{1}{\text{cos}}^2x \\ (\text{csc} x)' &= −\text{csc} x · \text{cot} x \\ (\text{sec} x)' &= \text{sec} x · \text{tan} x \\ (\text{cot} x)' &= −\text{csc}^2x \\ &\text{Inverse trigonometric derivatives}: \\ (\text{arcsin} x)' &= 1/\sqrt(1−x^2) \\ (\text{arccos} x)' &= −1/\sqrt(1−x^2) \\ (\text{arctan} x)' &= \frac{1}{1+x^2} \\ &\text{Examples}: \\ \to y &= \text{sin}(3x): y' = \text{cos}(3x) · 3 = 3cos(3x) (\text{via the chain rule}) \\ \to y &= \text{tan}(x^2): y' = \text{sec}^2(x^2) · 2x \end{aligned}$$

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Differentiating Exponential and Logarithmic Functions

Exponential derivatives:

(eˣ)’ = eˣ (its own derivative)

(aˣ)’ = aˣ · ln a (a > 0, a ≠ 1)

Logarithmic derivatives:

(ln x)’ = 1/x (x > 0)

(log_a x)’ = 1/(x · ln a)

Applying the chain rule:

(e^(f(x)))’ = e^(f(x)) · f’(x)

(ln(f(x)))’ = f’(x) / f(x)

Examples:

• y = e^(x²): y’ = e^(x²) · 2x
• y = ln(sin x): y’ = cos x / sin x = cot x
• y = e^(3x+1): y’ = 3e^(3x+1)

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Higher-Order Derivatives

The second derivative:

f”(x) = d²f/dx² = (f’(x))’

The nth derivative:

f⁽ⁿ⁾(x) = dⁿf/dxⁿ

Physical meaning:

• s(t): position
• s’(t) = v(t): velocity
• s”(t) = v’(t) = a(t): acceleration

Example:

• f(x) = x⁴
• f’(x) = 4x³
• f”(x) = 12x²
• f'''(x) = 24x
• f⁽⁴⁾(x) = 24
• f⁽ⁿ⁾(x) = 0 (n ≥ 5)

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Frequently Asked Questions

Q. How do I avoid mistakes applying the chain rule? A. Remember it as “outer derivative × inner derivative.” For y = f(g(x)), differentiate the outer function f while keeping g(x) intact, then multiply by the derivative of the inner function g(x). Practicing enough problems until it becomes automatic is the most effective approach.

Q. Do I need to memorize all the differentiation formulas? A. You should absolutely memorize the basic formulas (the power rule, sin/cos, eˣ, ln x). The quotient rule and chain rule can be re-derived from understanding the underlying logic rather than the formula itself. On an exam, though, memorizing the key formulas pays off in speed.