Home Class 11 Applied Maths Chapter 7: Calculus Exercise 7.3: Differentiation
Chapter 7 • Unit III

Exercise 7.3: Differentiation

Comprehensive guide and practice questions for CBSE Class 11 Applied Mathematics curriculum (2026-27).

7.3 Differentiation: First Principles, Geometric Slopes, and Rates of Change

In Applied Mathematics, corporate financial modeling, and engineering dynamics, the derivative quantifies instantaneous sensitivity: how rapidly an output metric (revenue, cost, production volume) responds to an infinitesimal change in an input variable. Geometrically, it defines the exact slope of the tangent line to a curve.

Mathematical Definition & The First Principle (Delta Method)

  • Definition of Derivative: Let \(f(x)\) be a real-valued function defined on an open interval. The derivative of \(f\) at \(x\), denoted by \(f'(x)\) or \(\frac{df}{dx}\), is defined as the limit: \[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \] provided this limit exists finitely.
  • Derivative at a Specific Point \(x = c\): \[ f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h} = \lim_{x \to c} \frac{f(x) - f(c)}{x - c} \]
  • Geometric Interpretation: The difference quotient \(\frac{f(x + h) - f(x)}{h}\) represents the slope of the secant line connecting points \(P(x, f(x))\) and \(Q(x + h, f(x + h))\). As \(h \to 0\), point \(Q\) travels along the curve toward \(P\), and the secant line approaches the tangent line at \(P\). \[ \text{Slope of Tangent at } (x_0, y_0) = m = f'(x_0) = \left. \frac{dy}{dx} \right|_{x = x_0} \]
  • One-Sided Derivatives (Differentiability Test): A function is differentiable at \(x = c\) if and only if its Left-Hand Derivative (LHD) equals its Right-Hand Derivative (RHD): \[ \text{LHD} = f'(c^-) = \lim_{h \to 0^+} \frac{f(c - h) - f(c)}{-h} \] \[ \text{RHD} = f'(c^+) = \lim_{h \to 0^+} \frac{f(c + h) - f(c)}{h} \] \[ f \text{ is differentiable at } c \iff \text{LHD} = \text{RHD} \]
  • The Fundamental Differentiability-Continuity Theorem: \[ \textbf{Differentiability} \implies \textbf{Continuity} \] If \(f(x)\) is differentiable at \(x = c\), then it is necessarily continuous at \(x = c\). Caution: The converse is FALSE. Continuity does not guarantee differentiability.

Standard First Principles Derivations (CBSE Delta Method)

Function \(f(x)\) Algebraic Step via First Principle Derivative \(f'(x)\)
Constant \(f(x) = c\) \(\lim_{h \to 0} \frac{c - c}{h} = \lim_{h \to 0} 0\) \(0\)
Power \(f(x) = x^n\) Binomial expansion of \((x + h)^n = x^n + n x^{n-1}h + \dots\) \(n x^{n-1}\)
Square Root \(f(x) = \sqrt{x}\) Rationalize: \(\frac{\sqrt{x+h} - \sqrt{x}}{h} \times \frac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h} + \sqrt{x}}\) \(\frac{1}{2\sqrt{x}}\)
Reciprocal \(f(x) = \frac{1}{x}\) Common denominator: \(\frac{x - (x+h)}{h x (x+h)} = \frac{-1}{x(x+h)}\) \(-\frac{1}{x^2}\)

Classic Example: The Non-Differentiable Continuous Function

Problem: Show that \(f(x) = |x|\) is continuous at \(x = 0\), but not differentiable at \(x = 0\).

Part 1: Proof of Continuity:

  • \(f(0) = |0| = 0\)
  • \(\lim_{x \to 0^-} |x| = \lim_{x \to 0^-} (-x) = 0\)
  • \(\lim_{x \to 0^+} |x| = \lim_{x \to 0^+} (x) = 0\)
Since \(\text{LHL} = \text{RHL} = f(0) = 0\), \(f(x)\) is continuous at \(x = 0\).

Part 2: Test of Differentiability at \(x = 0\):

  • Left-Hand Derivative (LHD): \[ \text{LHD} = \lim_{h \to 0^+} \frac{f(0 - h) - f(0)}{-h} = \lim_{h \to 0^+} \frac{|-h| - 0}{-h} = \lim_{h \to 0^+} \frac{h}{-h} = -1. \]
  • Right-Hand Derivative (RHD): \[ \text{RHD} = \lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0^+} \frac{|h| - 0}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1. \]

Since \(\text{LHD} = -1 \neq 1 = \text{RHD}\), the derivative at \(x = 0\) does not exist.

Geometric Meaning: The curve has a sharp corner (v-notch) at the origin with no unique tangent slope.

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