K2KARAKORAM
English
Open the app

State Board · Class 12 · Mathematics Part 2

Lesson 1: Differentiation

4 concepts · 32 practice questions with answers · Free to read

Chain rule: derivatives of composite functions

From daily life

A speedometer shows how fast the odometer changes; the odometer itself depends on engine turns. Rates multiply along the chain. Building on class 11 : you differentiated simple powers and trigonometric functions; now they sit inside one another.

Simple definition

If y = f(u) and u = g(x), the derivative of the composite function is dy/dx = (dy/du)·(du/dx). Work from the outside in, then multiply by the derivative of the inside.

Rule 1: Power of a function — d/dx [f(x)]ⁿ = n[f(x)]ⁿ⁻¹ · f′(x).

Rule 2: Exponential, log, trig — d/dx e^f = e^f · f′; d/dx log f = f′/f; d/dx sin f = cos f · f′; d/dx cos f = −sin f · f′.

How is the formula obtained? : y = (3x + 2)⁵: outside power 5(3x + 2)⁴, inside derivative 3, so dy/dx = 15(3x + 2)⁴; at x = 0 this is 15 · 16 = 240.

  • Easy : y = (2x + 1)³ ⇒ dy/dx = 3(2x + 1)² · 2 = 6(2x + 1)²; at x = 1 it is 54.
  • Medium : y = sin(x²) ⇒ dy/dx = cos(x²) · 2x; y = e^{x²} ⇒ 2x e^{x²}.
  • Exam-type : y = √(x² + 1) ⇒ dy/dx = x/√(x² + 1); at x = 1 it is 1/√2.
Remember

The derivative of the inside function is the factor people forget.

Common mistake

Writing d/dx sin(2x) = cos(2x) and forgetting the factor 2.

Shortcut

Name the inside u, differentiate the outside in u, multiply by du/dx. Memory clue : "Outside first, then times the inside."

CET trap

d/dx log(x² + 1) = 2x/(x² + 1), not 1/(x² + 1).

CET-style example : If y = e^{x²}, then dy/dx at x = 1 is (A) e (B) 2e (C) 2 (D) 1

Answer : (B). 2x e^{x²} = 2 · e. Time-saving method : Put x = 1 after differentiating.

Spot the difference

FunctionDerivative
(f(x))ⁿn (f(x))ⁿ⁻¹ f′(x)
e^{f(x)}e^{f(x)} f′(x)
log f(x)f′(x)/f(x)

Practice questions

Easy
  1. 1Differentiate (2x + 1)³ at x = 1.
  2. 2Differentiate cos(3x).
  3. 3Differentiate e^{2x + 1}.
Medium
  1. 4Differentiate sin(x²).
  2. 5Differentiate log(x² + 1) at x = 1.
  3. 6Differentiate √(x² + 1) at x = 1.
Challenging
  1. 7Differentiate (3x + 2)⁵ and find its value at x = 0.
  2. 8Differentiate e^{x²} at x = 1.
Show answers (try them yourself first)
  1. 16(2x + 1)² = 54.
  2. 2−3 sin 3x.
  3. 32e^{2x + 1}.
  4. 42x cos(x²).
  5. 52x/(x² + 1) = 1.
  6. 6x/√(x² + 1) = 1/√2 ≈ 0.707.
  7. 715(3x + 2)⁴ = 240.
  8. 82x e^{x²} = 2e ≈ 5.44.

Derivatives of inverse trigonometric functions

From daily life

A ramp's angle from its rise and run is an inverse trigonometric idea: the angle changes slowly when the ramp is flat and faster when it is steep.

Simple definition

For −1 < x < 1: d/dx sin⁻¹x = 1/√(1 − x²), d/dx cos⁻¹x = −1/√(1 − x²). For all x: d/dx tan⁻¹x = 1/(1 + x²), d/dx cot⁻¹x = −1/(1 + x²). If y = f⁻¹(x) then dy/dx = 1/(dx/dy).

Rule 1: With a chain — d/dx sin⁻¹(u) = u′/√(1 − u²); d/dx tan⁻¹(u) = u′/(1 + u²).

How is the formula obtained? : If y = sin⁻¹x then x = sin y, so dx/dy = cos y = √(1 − x²) and dy/dx = 1/√(1 − x²). At x = ½ this is 2/√3.

  • Easy : d/dx tan⁻¹x at x = 1 is 1/2.
  • Medium : d/dx tan⁻¹(2x) = 2/(1 + 4x²); at x = ½ it is 1.
  • Exam-type : d/dx sin⁻¹(x²) = 2x/√(1 − x⁴); at x = ½ it is 1/√(15/16) ≈ 1.033.
Remember

sin⁻¹ and cos⁻¹ derivatives are negatives of each other, and so are tan⁻¹ and cot⁻¹.

Common mistake

Forgetting the inner derivative: d/dx sin⁻¹(x²) is not 1/√(1 − x²).

Shortcut

Substitute a suitable trigonometric function to simplify before differentiating. Memory clue : "Co- functions carry a minus sign."

CET trap

1/√(1 − x²) needs |x| < 1; at x = ±1 the derivative does not exist.

CET-style example : d/dx tan⁻¹(2x) at x = ½ is (A) 1 (B) 2 (C) ½ (D) 4/5

Answer : (A). 2/(1 + 4·¼) = 2/2 = 1. Time-saving method : Put 2x = 1 inside first.

Spot the difference

FunctionDerivative
sin⁻¹x1/√(1 − x²)
cos⁻¹x−1/√(1 − x²)
tan⁻¹x1/(1 + x²)

Practice questions

Easy
  1. 1d/dx sin⁻¹x = ?
  2. 2d/dx cos⁻¹x = ?
  3. 3d/dx tan⁻¹x at x = 1?
Medium
  1. 4d/dx sin⁻¹x at x = ½?
  2. 5d/dx cos⁻¹x at x = ½?
  3. 6d/dx tan⁻¹(2x) at x = ½?
Challenging
  1. 7d/dx sin⁻¹(x²)?
  2. 8If y = tan⁻¹(x/3), find dy/dx.
Show answers (try them yourself first)
  1. 11/√(1 − x²).
  2. 2−1/√(1 − x²).
  3. 31/2.
  4. 42/√3 ≈ 1.155.
  5. 5−2/√3 ≈ −1.155.
  6. 61.
  7. 72x/√(1 − x⁴).
  8. 8(1/3)/(1 + x²/9) = 3/(9 + x²).

Implicit and logarithmic differentiation

From daily life

A circular track is described by x² + y² = 25; the track gives a relation between x and y but not y alone. We still need the slope at a point.

Simple definition

In an implicit function F(x, y) = 0 differentiate both sides with respect to x treating y as a function of x (d/dx y² = 2y dy/dx), then solve for dy/dx. For y = f(x)^{g(x)} take logarithms first (logarithmic differentiation).

Rule 1: Implicit — Differentiate each term, collect dy/dx terms on one side, solve.

Rule 2: Logarithmic — Take log of both sides: log y = g(x) log f(x), then differentiate; finally dy/dx = y × (that result).

How is the formula obtained? : y = xˣ ⇒ log y = x log x ⇒ (1/y)(dy/dx) = log x + 1 ⇒ dy/dx = xˣ(1 + log x). At x = 2 this is 4(1 + log 2) ≈ 6.77.

  • Easy : x² + y² = 25 at (3, 4): 2x + 2y y′ = 0 ⇒ y′ = −x/y = −3/4.
  • Medium : x² + xy + y² = 3 at (1, 1): 2x + y + x y′ + 2y y′ = 0 ⇒ 3 + 3y′ = 0 ⇒ y′ = −1.
  • Exam-type : y = xˣ at x = 1 gives dy/dx = 1·(1 + 0) = 1.
Remember

Every y-term gets a factor dy/dx.

Common mistake

Differentiating y² as 2y instead of 2y · dy/dx.

Shortcut

For products of many factors or variable exponents use logarithms. Memory clue : "Whenever y appears, dy/dx follows."

CET trap

Do not use the power rule on xˣ; the exponent also varies.

CET-style example : If x² + y² = 25, dy/dx at (3, 4) is (A) −3/4 (B) 3/4 (C) −4/3 (D) 4/3

Answer : (A). y′ = −x/y. Time-saving method : Use y′ = −x/y directly for circles.

Spot the difference

TypeFirst step
ImplicitDifferentiate both sides
Variable powerTake logarithm first

Practice questions

Easy
  1. 1Find dy/dx if x² + y² = 25 at (3, 4).
  2. 2Find dy/dx if x² + y² = 25 at (0, 5).
  3. 3d/dx of y² with respect to x?
Medium
  1. 4x² + xy + y² = 3: find dy/dx at (1, 1).
  2. 5xy = 12: find dy/dx at (3, 4).
  3. 6y = xˣ: find dy/dx.
Challenging
  1. 7Find dy/dx of y = xˣ at x = 2.
  2. 8y = x^{x²}? Find dy/dx.
Show answers (try them yourself first)
  1. 1−x/y = −3/4.
  2. 20.
  3. 32y dy/dx.
  4. 4−1.
  5. 5y + x y′ = 0 ⇒ y′ = −y/x = −4/3.
  6. 6xˣ(1 + log x).
  7. 74(1 + log 2) ≈ 6.77.
  8. 8log y = x² log x ⇒ y′ = x^{x²}(2x log x + x).

Parametric and higher-order derivatives

From daily life

A ball thrown in the air has x and y positions that both depend on time t. The path's slope comes from how fast y changes compared with x.

Simple definition

If x = f(t), y = g(t) are parametric equations, dy/dx = (dy/dt)/(dx/dt). The second derivative d²y/dx² = d/dx(dy/dx) = [d/dt(dy/dx)]/(dx/dt).

Rule 1: Slope — dy/dx = (dy/dt) ÷ (dx/dt), valid when dx/dt ≠ 0.

Rule 2: Second order — Differentiate dy/dx with respect to t, then divide by dx/dt again.

How is the formula obtained? : x = t², y = t³: dy/dx = 3t²/2t = 3t/2; d/dt(3t/2) = 3/2; d²y/dx² = (3/2)/(2t) = 3/(4t). At t = 2: dy/dx = 3 and d²y/dx² = 3/8.

  • Easy : y = x³ ⇒ y′ = 3x², y″ = 6x.
  • Medium : x = 2t, y = t² ⇒ dy/dx = 2t/2 = t.
  • Exam-type : x = t², y = t³ at t = 2: dy/dx = 3, d²y/dx² = 3/8.
Remember

d²y/dx² is not (dy/dx)², it is the derivative of dy/dx.

Common mistake

Dividing d²y/dt² by d²x/dt² to get d²y/dx².

Shortcut

Find dy/dx as a function of t first, then differentiate once more and divide by dx/dt. Memory clue : "Divide by dx/dt twice."

CET trap

Second derivative of a parametric curve needs the extra division by dx/dt.

CET-style example : x = t², y = t³: dy/dx at t = 2 is (A) 3 (B) 6 (C) 4 (D) 12

Answer : (A). 3t/2 = 3. Time-saving method : Cancel t before substituting.

Spot the difference

QuantityFormula
dy/dx(dy/dt)/(dx/dt)
d²y/dx²[d/dt(dy/dx)]/(dx/dt)

Practice questions

Easy
  1. 1y = x³: find d²y/dx².
  2. 2y = sin x: find y″.
  3. 3y = e^{2x}: find y″.
Medium
  1. 4x = 2t, y = t²: find dy/dx.
  2. 5x = t², y = t³: dy/dx at t = 2?
  3. 6x = t², y = t³: find d²y/dx².
Challenging
  1. 7x = t², y = t³: d²y/dx² at t = 2?
  2. 8If y = log x find y‴.
Show answers (try them yourself first)
  1. 16x.
  2. 2−sin x.
  3. 34e^{2x}.
  4. 4t.
  5. 53.
  6. 63/(4t).
  7. 73/8.
  8. 82/x³.