Is it correct to say: Applying chain rule but forgetting the inner derivative?

d/dx[f(g(x))] = f'(g(x))·g'(x). The factor g'(x) is often missed.

Is it correct to say: Using product rule (uv)' = u'v'?

Correct product rule: (uv)' = u'v + uv'. Two terms, cross-multiplied.

Is it correct to say: Computing d²y/dx² for parametric as (d²y/dt²)/(d²x/dt²)?

d²y/dx² = d/dx(dy/dx) = [d/dt(dy/dx)] × (dt/dx). Chain rule matters.

Is it correct to say: Forgetting to take log both sides in x^x-type problems?

For y = f(x)^{g(x)}: take log first, differentiate, then multiply by y.

Is it correct to say: Not converting log₁₀ to ln when using standard derivative?

d/dx(log_a x) = 1/(x ln a). For log₁₀, multiply by 1/ln 10 ≈ 0.4343.

Is it correct to say: Mixing up sin⁻¹ and 1/sin in derivative formulas?

d(sin⁻¹ x)/dx = 1/√(1−x²). NOT related to csc x = 1/sin x.

Differentiation

Derivatives of composite, implicit, parametric, logarithmic functions — Maharashtra HSC Maths Ch 3

📐 11 formulas✏️ 3 examples🎯 5 practice⚖️ 8 marks🏫 Maharashtra State Board📚 Class 12✓ 2025–26 syllabus

💡

Board Exam Tips

→Logarithmic differentiation of y = f(x)^g(x) type is a Maharashtra HSC 3-marker every year.
→Derivative of implicit function x² + y² = 25 or xy = k using dy/dx = −(∂F/∂x)/(∂F/∂y) is a 3-mark question.
→Parametric differentiation dy/dx = (dy/dt)/(dx/dt) is a 3-marker; state the formula before applying.
→Second-order derivative of a parametric function is a 4-mark HOTS problem — practise the chain rule carefully.
→For inverse trigonometric derivatives, memorise all six standard forms and their domains.

📐 Formulas(11)

Definition (first principle)

\frac{d y}{d x} = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}

Sum, Product, Quotient Rules★ Board fav

(u + v)^{'} = u^{'} + v^{'}; (uv)^{'} = u^{'} v + u v^{'}; (\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}

Chain Rule★ Board fav

\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)

Derivatives of Standard Functions★ Board fav

(x^{n})^{'} = n x^{n - 1}; (sin x)^{'} = cos x; (cos x)^{'} = - sin x; (e^{x})^{'} = e^{x}; (lo g x)^{'} = 1/ x

Inverse Trigonometric Derivatives

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

Logarithmic Differentiation★ Board fav

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

Implicit Differentiation★ Board fav

F (x, y) = 0 \Rightarrow \frac{d y}{d x} = - \frac{\partial F / \partial x}{\partial F / \partial y}

Parametric Differentiation★ Board fav

x = f (t), y = g (t) \Rightarrow \frac{d y}{d x} = \frac{d y / d t}{d x / d t}

Second-order Parametric Derivative

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

Higher-Order Derivatives

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

Logarithm of a Product

lo g (uv) = lo g u + lo g v, lo g (u / v) = lo g u - lo g v

✏️ Solved Examples

1Solved Exampleeasy3 steps

Find dy/dx if y = sin(x²).

Chain rule: outer sin, inner x².

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

2Solved Exampleboard3 steps

If y = xˣ, find dy/dx.

Take log both sides.

ln y = x ln x

3Solved ExampleHOTS5 steps

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

First derivatives with respect to θ.

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

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Applying chain rule but forgetting the inner derivative
✓d/dx[f(g(x))] = f'(g(x))·g'(x). The factor g'(x) is often missed.
2
Using product rule (uv)' = u'v'
✓Correct product rule: (uv)' = u'v + uv'. Two terms, cross-multiplied.
3
Computing d²y/dx² for parametric as (d²y/dt²)/(d²x/dt²)
✓d²y/dx² = d/dx(dy/dx) = [d/dt(dy/dx)] × (dt/dx). Chain rule matters.
4
Forgetting to take log both sides in x^x-type problems
✓For y = f(x)^{g(x)}: take log first, differentiate, then multiply by y.
5
Not converting log₁₀ to ln when using standard derivative
✓d/dx(log_a x) = 1/(x ln a). For log₁₀, multiply by 1/ln 10 ≈ 0.4343.
6
Mixing up sin⁻¹ and 1/sin in derivative formulas
✓d(sin⁻¹ x)/dx = 1/√(1−x²). NOT related to csc x = 1/sin x.

🎯 Practice Yourself

🎯Practice Yourself5 questions

Q1
Find dy/dx if y = tan⁻¹(x/√(1−x²)).
Q2
If x² + y² = 25, find dy/dx at (3, 4).
Q3
Differentiate log(sin x) with respect to x.
Q4
If y = e^(x²), find d²y/dx².
Q5
Find dy/dx if x = a cos³t, y = a sin³t.

📝 Notes

Differentiation — Maharashtra HSC Overview

Chapter 3 of the Maharashtra Class 12 Mathematics (Balbharati) textbook is a heavy scoring chapter — differentiation of composite, implicit, parametric, and logarithmic functions.

Maharashtra syllabus specifics

Logarithmic differentiation is a Balbharati-heavy topic; expect at least one 3- or 4-mark question every HSC year.
Parametric differentiation and its second-order derivative are two separate subtopics — both asked.
Derivative of inverse trigonometric functions using substitutions (e.g. x = tan θ) is asked with a rider "using the substitution show that…" for 3 marks.
First principles proof (derivative from limit definition) for standard functions like sin x, cos x, e^x is a Maharashtra Board specific 3-mark question — CBSE rarely asks this.

Strategy

Always write down the formula BEFORE substituting. Formula shown = 1 mark guaranteed.
For x^x and similar: take log, differentiate, then multiply by y.
For parametric second-order, take extra care with chain rule dt/dx = 1/(dx/dt).
Practise implicit differentiation on conics — direct HSC application in trajectory numericals.