What this chapter is about
This chapter builds the theoretical foundation for calculus by formalising two connected ideas: continuity (a function has no breaks or jumps) and differentiability (a function has a well-defined slope at each point). In Class 11 you met limits and the idea of a derivative; now you study when derivatives exist and how to compute them for a wide range of functions.
You will learn precise definitions of continuity and differentiability, then develop powerful differentiation techniques: the chain rule for composite functions, derivatives of implicit relations, logarithmic differentiation for complicated products and powers, and derivatives of parametric and second-order functions. The chapter also introduces the mean value theorems, which connect a function's average rate of change to its instantaneous rate.
After completing this chapter you should be able to test whether a given function is continuous or differentiable at a point, differentiate functions that involve compositions, implicit relations, inverse trigonometric forms, and exponential or logarithmic expressions, and state and apply Rolle's theorem and the Mean Value Theorem.
Key ideas
- Continuity at a point: A function f is continuous at x = a when lim x→a f(x) exists, f(a) is defined, and lim x→a f(x) = f(a). Informally, you can draw the curve through a without lifting your pen.
- Differentiability at a point: f is differentiable at x = a when lim h→0 [f(a + h) − f(a)] / h exists and is finite. This limit, if it exists, is the derivative f ′(a).
- Differentiability implies continuity: If f is differentiable at a point, it must be continuous there; the converse is false (a sharp corner can be continuous but not differentiable).
- Chain rule: For a composite y = f(g(x)), the derivative is dy/dx = f ′(g(x)) · g ′(x). In Leibniz form, dy/dx = (dy/du)(du/dx) where u = g(x).
- Implicit differentiation: When y is not isolated (e.g., x² + y² = 25), differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.
- Logarithmic differentiation: For products, quotients or variable exponents, take ln of both sides, differentiate, then multiply back by the original function.
- Parametric differentiation: If x = φ(t) and y = ψ(t), then dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0.
- Mean Value Theorem (MVT): If f is continuous on [a, b] and differentiable on (a, b), there exists some c in (a, b) with f ′(c) = [f(b) − f(a)] / (b − a). Rolle's theorem is the special case when f(a) = f(b), giving f ′(c) = 0.
Formulas and facts to remember
- Formula / Rule: d/dx (xⁿ) = n x^(n−1) · Meaning: Power rule, valid for any real n.
- Formula / Rule: d/dx (eˣ) = eˣ · Meaning: Exponential function is its own derivative.
- Formula / Rule: d/dx (aˣ) = aˣ ln a · Meaning: Base-a exponential, a > 0, a ≠ 1.
- Formula / Rule: d/dx (ln x) = 1/x · Meaning: Natural logarithm, x > 0.
- Formula / Rule: d/dx (sin x) = cos x ; d/dx (cos x) = −sin x · Meaning: Basic trigonometric derivatives.
- Formula / Rule: d/dx (tan x) = sec² x ; d/dx (cot x) = −cosec² x · Meaning:
- Formula / Rule: d/dx (sin⁻¹ x) = 1/√(1 − x²) ; d/dx (cos⁻¹ x) = −1/√(1 − x²) · Meaning: Inverse sine and cosine, |x| < 1.
- Formula / Rule: d/dx (tan⁻¹ x) = 1/(1 + x²) · Meaning: Inverse tangent, all real x.
- Formula / Rule: Product rule: d/dx (uv) = u (dv/dx) + v (du/dx) · Meaning:
- Formula / Rule: Quotient rule: d/dx (u/v) = [v (du/dx) − u (dv/dx)] / v² · Meaning:
- Formula / Rule: Second derivative: d²y/dx² = d/dx (dy/dx) · Meaning: Measures rate of change of the slope.
Worked examples
Example 1 — Testing continuity
Problem: Examine whether f(x) = |x − 2| is continuous at x = 2.
Solution:
For x ≥ 2, |x − 2| = x − 2; for x < 2, |x − 2| = 2 − x.
Left-hand limit: lim x→2⁻ (2 − x) = 0.
Right-hand limit: lim x→2⁺ (x − 2) = 0.
Value at x = 2: f(2) = |0| = 0.
Since lim x→2 f(x) = 0 = f(2), the function is continuous at x = 2.
Example 2 — Implicit differentiation
Problem: If x² + y² = 49, find dy/dx.
Solution:
Differentiate both sides with respect to x:
2x + 2y (dy/dx) = 0.
Solve for dy/dx:
dy/dx = −x / y (provided y ≠ 0).
At the point (7, 0) the derivative is undefined, matching the vertical tangent there.
Example 3 — Logarithmic differentiation
Problem: Differentiate y = x^(sin x), where x > 0.
Solution:
Take natural logarithm: ln y = sin x · ln x.
Differentiate both sides with respect to x:
(1/y)(dy/dx) = cos x · ln x + sin x · (1/x).
Multiply both sides by y:
dy/dx = x^(sin x) [cos x · ln x + (sin x)/x].
Common mistakes
- Forgetting to check that f(a) is defined when testing continuity → always verify the function value exists at the point.
- Assuming continuity guarantees differentiability → remember f(x) = |x| is continuous at 0 but has no derivative there.
- Omitting the inner derivative in the chain rule → when differentiating sin(3x), the answer is 3 cos(3x), not cos(3x).
- Treating y as a constant in implicit differentiation → every time you differentiate a term containing y, multiply by dy/dx.
- Confusing the Mean Value Theorem with Rolle's theorem → MVT applies when f(a) ≠ f(b); Rolle's requires f(a) = f(b).
Quick revision
- Continuous at a means the limit equals the function value there.
- Differentiable at a means the limit of the difference quotient exists; this forces continuity but not vice versa.
- Chain rule: multiply the outer derivative by the inner derivative.
- Implicit differentiation: differentiate the whole equation, then isolate dy/dx.
- Logarithmic differentiation turns products and powers into sums, simplifying differentiation.
- MVT guarantees at least one point where instantaneous rate equals average rate over an interval.