Is it correct to say: Writing (uv)' = u'v' instead of the product rule?

Product rule is u'v + uv'. Two terms, not one. Practise on y = x sin x to lock it in.

Is it correct to say: Forgetting the inner derivative in chain rule, e.g. d/dx sin(3x) = cos(3x)?

Chain rule: d/dx sin(3x) = cos(3x) · 3 = 3 cos(3x). Always multiply by derivative of the inner function.

Is it correct to say: Applying the power rule to x^x as x · x^{x-1}?

Power rule needs a CONSTANT exponent. For x^x take logs first: ln y = x ln x, then differentiate implicitly.

Is it correct to say: Writing (cos x)' = sin x (missing minus sign)?

(cos x)' = −sin x. The MINUS is on cos, cot, cosec and their inverses (cos⁻¹, cot⁻¹, cosec⁻¹).

Is it correct to say: For parametric x(t), y(t) writing dy/dx = dy/dt × dx/dt?

dy/dx = (dy/dt) ÷ (dx/dt). It is a quotient. Multiplication gives a dimensionally wrong answer.

Is it correct to say: Testing continuity by checking only that f(a) is defined?

Continuity needs LHL = RHL = f(a), all three equal and finite. Missing any check loses marks.

Continuity and Differentiability

Continuity, differentiability, chain rule, product rule, quotient rule, implicit and logarithmic differentiation — NCERT Class 12 Maths Ch 5

📐 14 formulas✏️ 3 examples🎯 6 practice⚖️ 6-8 marks🏫 CBSE📚 Class 12✓ 2025–26 syllabus

💡

Board Exam Tips

→Chain rule and product rule together carry a 4-mark question almost every year.
→Logarithmic differentiation is the go-to trick when you see a function raised to a function, e.g. x^x or (sin x)^{cos x}.
→Continuity of a piecewise function at a boundary point is a 3-mark favourite — check LHL, RHL and f(a) all three.
→For parametric forms remember dy/dx = (dy/dt)/(dx/dt), not dy/dt · dx/dt.
→Always simplify after differentiating — CBSE gives 1 mark for the final clean form.

📐 Formulas(14)

Definition of Derivative

f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}

Continuity at a Point★ Board fav

lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x) = f (a)

Differentiability implies Continuity

f differentiable at a \Rightarrow f continuous at a

Sum, Difference, Constant Multiple

(u \pm v)^{'} = u^{'} \pm v^{'}, (c u)^{'} = c u^{'}

Product Rule★ Board fav

\frac{d}{d x} (uv) = u^{'} v + u v^{'}

Quotient Rule★ Board fav

\frac{d}{d x} (\frac{u}{v}) = \frac{u ^{'} v - u v ^{'}}{v ^{2}}

Chain Rule★ Board fav

\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}

Standard Derivatives (Trig)★ Board fav

(sin x)^{'} = cos x, (cos x)^{'} = - sin x, (tan x)^{'} = sec^{2} x

Standard Derivatives (Inverse Trig)★ Board fav

(sin^{- 1} x)^{'} = \frac{1}{1 - x ^{2}}, (tan^{- 1} x)^{'} = \frac{1}{1 + x ^{2}}

Exponential and Logarithm

(e^{x})^{'} = e^{x}, (a^{x})^{'} = a^{x} ln a, (ln x)^{'} = \frac{1}{x}

Logarithmic Differentiation★ Board fav

y = [f (x)]^{g (x)} \Rightarrow ln y = g (x) ln f (x)

Parametric Differentiation★ Board fav

\frac{d y}{d x} = \frac{d y / d t}{d x / d t}

Second Derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{d y}{d x})

Implicit Differentiation

\frac{d}{d x} (y^{n}) = n y^{n - 1} \frac{d y}{d x}

✏️ Solved Examples

1Solved Exampleeasy3 steps

Differentiate y = (2x + 3)^5 with respect to x.

Identify outer function (·)^5 and inner function 2x+3. Apply chain rule.

\frac{d y}{d x} = 5 (2 x + 3)^{4} \cdot \frac{d}{d x} (2 x + 3)

2Solved Exampleboard3 steps

If y = x^{\sin x}, find dy/dx.

Base and exponent are both functions of x — use logarithmic differentiation.

ln y = sin x \cdot ln x

3Solved ExampleHOTS5 steps

If x = a(θ − sin θ), y = a(1 − cos θ), find d²y/dx² at θ = π/2.

Compute dx/dθ and dy/dθ.

\frac{d x}{d θ} = a (1 - cos θ), \frac{d y}{d θ} = a sin θ

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Writing (uv)' = u'v' instead of the product rule
✓Product rule is u'v + uv'. Two terms, not one. Practise on y = x sin x to lock it in.
2
Forgetting the inner derivative in chain rule, e.g. d/dx sin(3x) = cos(3x)
✓Chain rule: d/dx sin(3x) = cos(3x) · 3 = 3 cos(3x). Always multiply by derivative of the inner function.
3
Applying the power rule to x^x as x · x^{x-1}
✓Power rule needs a CONSTANT exponent. For x^x take logs first: ln y = x ln x, then differentiate implicitly.
4
Writing (cos x)' = sin x (missing minus sign)
✓(cos x)' = −sin x. The MINUS is on cos, cot, cosec and their inverses (cos⁻¹, cot⁻¹, cosec⁻¹).
5
For parametric x(t), y(t) writing dy/dx = dy/dt × dx/dt
✓dy/dx = (dy/dt) ÷ (dx/dt). It is a quotient. Multiplication gives a dimensionally wrong answer.
6
Testing continuity by checking only that f(a) is defined
✓Continuity needs LHL = RHL = f(a), all three equal and finite. Missing any check loses marks.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Differentiate y = sin(cos(x²)) with respect to x.
Q2
If y = (log x)^x, find dy/dx.
Q3
For what value of k is f(x) = kx + 1 for x ≤ 5, f(x) = 3x − 5 for x > 5 continuous at x = 5?
Q4
If x^y = e^{x-y}, find dy/dx.
Q5
If y = tan⁻¹((sin x)/(1 + cos x)), find dy/dx.
Q6
If x = a cos³ θ, y = a sin³ θ, find dy/dx at θ = π/4.

📝 Notes

Continuity and Differentiability

Chapter 5 formalises the idea of a "smooth" function and gives the algebraic machinery — product, quotient, chain, logarithmic and implicit differentiation — that runs through the rest of Class 12 calculus.

Continuity vs Differentiability

Continuous at a means the graph has no jump, no hole and no vertical asymptote at a.
Differentiable at a means the graph has a well-defined tangent (no corner, cusp or vertical tangent).
Differentiable ⇒ continuous. The converse fails: think of y = |x| at the origin.

The three-line continuity test

For f to be continuous at x = a, all three must exist and be equal:

LHL: limit as x approaches a from the left
RHL: limit as x approaches a from the right
f(a) itself

A single failure — for instance LHL ≠ RHL, or f(a) undefined — makes f discontinuous.

Choosing a differentiation tool

Polynomial or standard function — apply the direct formula.
Function of a function — chain rule.
Product of two functions — product rule.
Quotient of two functions — quotient rule (or rewrite as product with negative power).
Variable base AND variable exponent — logarithmic differentiation.
Equation not solved for y — implicit differentiation.
x and y both given in terms of a parameter t — parametric formula.

Sanity checks

Derivative of a constant is 0 — always.
If you get a huge, ugly expression, try factoring or applying an identity before you write "final answer".
The derivative of an even function is odd, and vice versa. Use this as a mistake radar.