NCERT Solutions for Class 12 Maths Chapter 5 – Continuity and Differentiability Miscellaneous Exercise

Flashcard for Question 1

Quick Tip: Use the chain rule – derivative of (u)^n is n(u)^(n−1) × (du/dx). Here, u = (3x² − 9x + 5).

Flashcard for Question 2

Quick Tip: Apply the chain rule + power rule separately to each term: for sin³x use 3sin²x·cosx, and for cos⁶x use 6cos⁵x·(−sinx).

Flashcard for Question 3

Quick Tip: Use the product rule: d(uv)/dx = u′v + uv′, with u = (5x)³ and v = cos(2x). Don’t forget the chain rule when differentiating cos(2x).

Flashcard for Question 4

Quick Tip: Use the derivative of sin⁻¹(u) = 1/√(1−u²) × du/dx, where u = x√x = x^(3/2). Be careful with the chain rule on x^(3/2).

Flashcard for Question 5

Quick Tip: Use the quotient rule with u = cos⁻¹(x/2) and v = √(2x+7). For u′, recall d/dx[cos⁻¹(u)] = −1/√(1−u²) × u′. For v′, apply chain rule on (2x+7)^(1/2).

Flashcard for Question 6

Quick Tip: First simplify the inside using trig identities (it reduces neatly if you rationalize). Then apply derivative of cot⁻¹(u) = −1/(1+u²) · du/dx.

👉 Common mistake: Jumping into differentiation without simplification — it makes the problem unnecessarily messy.

Flashcard for Question 7

Quick Tip: Use logarithmic differentiation — take ln(y) to bring the exponent down: ln(y) = logx · ln(logx), then differentiate both sides.

👉 Common mistake: Forgetting to apply the product rule when differentiating logx · ln(logx).

Flashcard for Question 8

Quick Tip: Apply the chain rule: derivative of cos(u) is −sin(u)·u′, with u = a cosx + b sinx.

Common Mistake: Forgetting that d/dx(cosx) = −sinx and d/dx(sinx) = cosx — both signs matter!

Exam Insight: These questions test your ability to combine chain rule + linear combinations of trig functions. Writing u = a cosx + b sinx first keeps the work clean and avoids sign errors.

Flashcard for Question 10

Quick Tip: Break it into terms. For x^x, use logarithmic differentiation). For x^a apply power rule. For a^x, use exponential rule. a^a is constant → derivative 0.

Common Mistake: Forgetting that x^x needs special handling (not the simple power rule). Also, many students mistakenly differentiate a^a even though it’s constant.

Exam Insight: When you see mixed bases and exponents, check carefully which parts depend on x. This style of question is a favourite in exams to test if you know when to use logarithmic differentiation

Flashcard for Question 12

Quick Tip: Since both x and y are in terms of t, use parametric differentiation.

Common Mistake: Forgetting to divide dy/dt by dx/dt, or mixing up signs when differentiating -cos t and -sin t.

Exam Insight: Parametric form questions are common – examiners want to see if you recall the formula dy/dx = (dy/dt)/(dx/dt). If you write derivatives clearly before dividing, you minimize sign errors and secure full marks.

Flashcard for Question 13

Quick Tip: Use derivative of sin⁻¹(u) = 1/√(1-u²) · du/dx for each term. For the second part, apply chain rule carefully with u = √(1-x²). Simplify at the end – it reduces neatly.

Flashcard for Question 14

Quick Tip: Differentiate implicitly using product and chain rules:
d/dx[x√(1+y)] + d/dx[y√(1+x)] = 0,
then collect terms containing dy/dx and solve for dy/dx to get −1/(1+x)^2.

Common Mistake: Forgetting that d/dx[√(1+y)] = (1/(2√(1+y)))·dy/dx (i.e. missing the dy/dx factor) or dropping the 1/2 factors from chain rule during algebra.

Exam Insight: After differentiating, group all dy/dx terms on one side and factor them out before simplifying — this clean algebra step usually secures full method marks and avoids sign errors.

Flashcard for Question 16

Quick Tip: Differentiate implicitly: sin y·(dy/dx) = cos(a+y) + x·(-sin(a+y))·(dy/dx). Collect dy/dx terms to get (x sin(a+y) − sin y)·(dy/dx) = cos(a+y). Use the identity sin a = sin(a+y)cos y − cos(a+y)sin y and substitute cos y = x cos(a+y) to simplify the denominator to sin a / cos(a+y). Hence dy/dx = cos^2(a+y)/sin a.

Common Mistake: Forgetting to use cos y = x cos(a+y) inside the trigonometric identity (so you can express x sin(a+y) − sin y in terms of sin a), or making sign errors when moving dy/dx terms to one side.

Exam Insight: Write the intermediate step (x sin(a+y) − sin y)·(dy/dx) = cos(a+y) and then show the short identity manipulation: sin a = sin(a+y)cos y − cos(a+y)sin y = cos(a+y)(x sin(a+y) − sin y). This justifies the final simplification cleanly. Also note cos a ≠ ±1 ensures sin a ≠ 0 so the division is valid.

Flashcard for Question 17

Quick Tip: For second derivative in parametric form, use d²y/dx² = (d/dt(dy/dx)) / (dx/dt). Always find dy/dt and dx/dt first, simplify, then differentiate again.

Flashcard for Question 19

Quick Tip: Start with sin(A+B) = sinA cosB + cosA sinB, then differentiate both sides w.r.t. A. You’ll get cos(A+B) on the left and the sum formula for cosines on the right.

Common Mistake: Forgetting that derivative of sinA is cosA and derivative of cosA is −sinA, which leads to wrong signs in the final formula.

Exam Insight: This is a standard derivation — examiners want to see if you can connect trigonometric identities with calculus. Writing the differentiation step clearly is enough to earn full marks.