Derivatives — Limit Definition, First Principles & Differentiation Rules

Acceptance Standard: This chapter serves as a stand-alone textbook for any beginner studying IIT Madras BSMA1001 (Mathematics for Data Science I). It develops differential calculus from first principles, bridges geometry with symbolic computation, proves the fundamental differentiation rules, and dissects all eight recurring exam question patterns.

1. Orientation & Mental Model: The Geometry of Instantaneous Change

The Tangent Line Problem

        y ^
          |                 * (x + h, f(x + h))
          |                /
          |        *      /  <-- Secant line slope = [f(x+h) - f(x)] / h
          |       /      /
          |      * (x, f(x))
          |     /
          +----+--------+-------------> x
               x       x + h

The Limiting Transition: Secant to Tangent

1. The point $(x + h, f(x + h))$ slides smoothly along the curve toward $(x, f(x))$.
2. The secant line pivots around $(x, f(x))$.
3. At the exact boundary limit $h \to 0$, the secant line freezes into the tangent line—a unique line that grazes the curve at $(x, f(x))$, matching its instantaneous trajectory.

2. Terminology, Notation & Etymology

3. Foundational Mechanics: Formal Definition & First Principles

The Definition of the Derivative

Deriving Standard Derivatives from First Principles

1. The Power Rule: $f(x) = x^n$ (for $n \in \mathbb{N}$)

2. The Exponential Function: $f(x) = e^x$

3. The Natural Logarithm: $f(x) = \ln x$ (for $x > 0$)

4. Operational Rules: Product, Quotient & Chain Rules

Proof of the Product Rule (The Add-and-Subtract Trick)

5. Failure Modes & Common Exam Pitfalls

         The Non-Negotiable Hierarchy:
             Differentiability
                   │
                   ▼ (Guaranteed)
               Continuity
                   │
                   ▼ (NOT reversible!)
            Does NOT imply Differentiability!

Pitfall 1: Continuity Does NOT Guarantee Differentiability

• Continuous at $0$: $\lim_{x \to 0} |x| = 0 = f(0)$.
• Left-hand derivative: $\lim_{h \to 0^-} \frac{|0 + h| - 0}{h} = \lim_{h \to 0^-} \frac{-h}{h} = \mathbf{-1}$.
• Right-hand derivative: $\lim_{h \to 0^+} \frac{|0 + h| - 0}{h} = \lim_{h \to 0^+} \frac{h}{h} = \mathbf{+1}$.
• Because the left and right limits do not match, $f'(0)$ does not exist. Graphically, this is a sharp corner (cusp).

Pitfall 2: The Chain Rule "Forgotten Tail"

• Correct: $\frac{1}{3x^2 + 5} \times \frac{d}{dx}(3x^2 + 5) = \frac{6x}{3x^2 + 5}$.
• Common blunder: Writing $\frac{1}{3x^2 + 5}$, forgetting to multiply by the derivative of the inside $g'(x) = 6x$.

Pitfall 3: Quotient Rule Numerator Sign Inversion

• Correct: $\frac{f'g - fg'}{g^2}$ ("Low d-High minus High d-Low over Low squared").
• Blunder: $\frac{fg' - f'g}{g^2}$. This flips the sign of the entire derivative, turning positive slopes negative!

6. Question-Pattern Anatomy (8 Finite Exam Shapes)

Shape 1: First Principles Limit Recognition

• Cue: "Find the value of $\lim_{h \to 0} \frac{\sin(x + h) - \sin(x)}{h}$."
• Attack: Do not expand trigonometry with angle-sum formulas! Recognize this as the exact limit definition of $\frac{d}{dx}[\sin x] = \cos x$.

Shape 2: Differentiability at Piecewise Junctions

• Cue: "For what values of $a$ and $b$ is $f(x) = \begin{cases} x^2 & x \le 1 \\ ax + b & x > 1 \end{cases}$ differentiable everywhere?"
• Attack:
  1. Continuity at $x = 1$: $1^2 = a(1) + b \implies a + b = 1$.
  2. Matching derivatives at $x = 1$: $\left.\frac{d}{dx}(x^2)\right|_{x=1} = 2(1) = 2$. Right derivative is $a$. Thus $a = 2$.
  3. Solve for $b$: $2 + b = 1 \implies b = -1$.

Shape 3: Equation of Tangent and Normal Lines

• Cue: "Find the equation of the normal line to $y = x^3 - 2x + 4$ at $x = 1$."
• Attack:
  1. Point: $y(1) = 1 - 2 + 4 = 3 \implies (1, 3)$.
  2. Derivative: $y' = 3x^2 - 2 \implies m_{\text{tangent}} = 3(1)^2 - 2 = 1$.
  3. Normal slope: $m_{\text{normal}} = -\frac{1}{m_{\text{tangent}}} = -1$.
  4. Equation: $y - 3 = -1(x - 1) \implies y = -x + 4$.

Shape 4: Repeated Chain Rule with Trig & Exponentials

• Cue: "Find $\frac{d}{dx}[e^{\sin(x^2)}]$."
• Attack: Work from outside to inside: $\frac{d}{dx}[e^u] = e^u \cdot u' = e^{\sin(x^2)} \cdot \cos(x^2) \cdot 2x = \mathbf{2x \cos(x^2) e^{\sin(x^2)}}$

Shape 5: Logarithmic Differentiation for Variable Exponents ($f(x)^{g(x)}$)

• Cue: "Differentiate $y = x^x$."
• Attack:
  1. Take natural log: $\ln y = x \ln x$.
  2. Implicit differentiation: $\frac{1}{y} y' = 1 \cdot \ln x + x \cdot \frac{1}{x} = \ln x + 1$.
  3. Multiply by $y$: $y' = x^x (\ln x + 1)$.

Shape 6: Linear Approximation via Differentials

• Cue: "Use linear approximation to estimate $\sqrt{4.08}$."
• Attack:
  1. $f(x) = \sqrt{x}$, anchor point $a = 4$, $\Delta x = 0.08$.
  2. $f(4) = 2$, $f'(x) = \frac{1}{2\sqrt{x}} \implies f'(4) = \frac{1}{4} = 0.25$.
  3. $L(4.08) = f(4) + f'(4)\Delta x = 2 + 0.25(0.08) = \mathbf{2.02}$.

Shape 7: Evaluating Indeterminate Limits via L'Hôpital's Rule

• Cue: "Evaluate $\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}$."
• Attack: Form is $\frac{0}{0}$. Differentiate top and bottom: $\lim_{x \to 0} \frac{e^x - 1}{2x} \quad \left(\frac{0}{0}\right) \implies \lim_{x \to 0} \frac{e^x}{2} = \mathbf{\frac{1}{2}}$

Shape 8: Corner and Vertical Tangent Testing

• Cue: "Is $f(x) = x^{1/3}$ differentiable at $x = 0$?"
• Attack: $f'(0) = \lim_{h \to 0} \frac{h^{1/3} - 0}{h} = \lim_{h \to 0} \frac{1}{h^{2/3}} = +\infty$ The limit is infinite. Thus, the curve has a vertical tangent, and $f$ is not differentiable at $0$.

7. Interactive Verification & Attempt-First Self-Checks

8. Next Action & Spacing Rail

• Next Invariant Chapter: Proceed to Applications of Derivatives — Maxima, Minima & Tangents to use first and second derivative tests for curve sketching and optimization.
• Data Science Optimization Connection: In BSMA1003 (Maths 2), you will extend this 1D derivative into partial derivatives and the gradient vector $\nabla f$, the mathematical foundation of training modern neural networks.
• Flashcard Prompt: What are the three non-differentiable scenarios? (Sharp corner, vertical tangent, discontinuity).