StudyRaah StudyRaah XI-XIIOpen StudyRaah

Maharashtra State Board · HSC Commerce · Class 12 (Std XII)

Class 12 Maths and Statistics Part 1 Chapter 3: Differentiation

Short notes, key concepts, quick revision and practice MCQs with answers

Building on Std XI, this chapter extends differentiation to composite functions, inverse functions, logarithmic differentiation, implicit and parametric functions, and derivatives of the second order.

Key concepts: Differentiation

Composite functions (chain rule)
If y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx). For example d/dx (x² + 1)⁵ = 5(x² + 1)⁴ × 2x.
Standard results by chain rule
d/dx [f(x)]ⁿ = n[f(x)]ⁿ⁻¹ f′(x); d/dx e^f(x) = e^f(x) f′(x); d/dx log f(x) = f′(x)/f(x); d/dx a^f(x) = a^f(x) log a · f′(x).
Inverse functions
If y = f(x) has an inverse x = f⁻¹(y), then dx/dy = 1/(dy/dx), provided dy/dx ≠ 0.
Logarithmic differentiation
For functions like u^v or products and quotients of many factors, take log on both sides, differentiate and multiply by y. E.g. d/dx (xˣ) = xˣ(1 + log x).
Implicit functions
When x and y are mixed, as in x² + y² = 25, differentiate every term with respect to x treating y as a function of x, then solve for dy/dx.
Parametric functions
If x = f(t) and y = g(t), then dy/dx = (dy/dt)/(dx/dt), dx/dt ≠ 0.
Second order derivative
d²y/dx² = d/dx (dy/dx), also written f″(x) or y₂.

Quick revision: important points

Practice MCQs with answers

  1. d/dx (x² + 1)⁵ equals:

    1. 5(x² + 1)⁴
    2. 10x(x² + 1)⁴
    3. 10x(x² + 1)⁵
    4. 2x(x² + 1)⁴
    Show answer

    Answer: B. 10x(x² + 1)⁴
    Chain rule: 5(x² + 1)⁴ × 2x.

  2. d/dx e^(3x) equals:

    1. e^(3x)
    2. 3e^(3x)
    3. e^(3x)/3
    4. 3x e^(3x − 1)
    Show answer

    Answer: B. 3e^(3x)
    d/dx e^(ax) = a e^(ax).

  3. d/dx log(2x + 3) equals:

    1. 1/(2x + 3)
    2. 2/(2x + 3)
    3. 2x + 3
    4. log 2
    Show answer

    Answer: B. 2/(2x + 3)
    f′(x)/f(x) = 2/(2x + 3).

  4. d/dx √(x² + 1) equals:

    1. 1/(2√(x² + 1))
    2. x/√(x² + 1)
    3. 2x √(x² + 1)
    4. x/(x² + 1)
    Show answer

    Answer: B. x/√(x² + 1)
    2x/(2√(x² + 1)).

  5. If y = f(x) and x = f⁻¹(y), then dx/dy equals:

    1. dy/dx
    2. 1/(dy/dx)
    3. −dy/dx
    4. (dy/dx)²
    Show answer

    Answer: B. 1/(dy/dx)
    Derivative of the inverse function.

All Class 12 Maths and Statistics chapters

Class 12 Maths and Statistics notes: all chapters