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27 July 2026

📖 Maths Important Formulas

📚 Most Important Mathematics Formulas for Quick Revision & Exam Preparation

📐 Calculus Formula Sheet

🚀 Student Sathidar Edition Designed for B.E. Sem 1 & 2 (GTU) – Complete Formula Reference

🔍 What is Calculus? Calculus is the mathematical study of continuous change. In engineering, it helps us analyze circuits, forces, motion, and rates of change.

🎯 Who is this for? First-year engineering students following the GTU (Gujarat Technological University) syllabus.

📌 What’s inside? All standard Differentiation & Integration formulas, GTU-focused star ratings, common mistakes, exam tips, quick revision box, and FAQ.

📐 Differentiation 📊 Integration ⚠️ Common Mistakes 🎯 Exam Tips ⏱️ Quick Revision ❓ FAQ

📐 Differentiation Formulas

Basic Rules

#	Formula	Name / Rule	GTU Focus
1	d/dx (k) = 0	Constant Rule	⭐
2	d/dx (x) = 1	Identity Rule	⭐
3	d/dx (1) = 0	Derivative of 1	—
4	d/dx [f(x) ± g(x)] = f'(x) ± g'(x)	Sum / Difference	—
5	d/dx [k·f(x)] = k·f'(x)	Constant Multiple	—
6	d/dx [f(x)·g(x)] = f·g' + g·f'	Product Rule	⭐
7	d/dx [f/g] = (g·f' − f·g') / g²	Quotient Rule	⭐
8	d/dx f(g(x)) = f'(g)·g'	Chain Rule	⭐⭐⭐

Power & Standard Functions

⭐ Most Important – Power Rule

d/dx (xⁿ) = n·xⁿ⁻¹

#	Formula	Name	GTU Focus
9	d/dx (xⁿ) = n·xⁿ⁻¹	Power Rule	⭐⭐⭐
10	d/dx (1/x) = −1/x²	Reciprocal	⭐
11	d/dx (√x) = 1/(2√x)	Square Root	⭐
12	d/dx (eˣ) = eˣ	Exponential (e)	⭐⭐
13	d/dx (aˣ) = aˣ · ln a	Exponential (a)	—
14	d/dx (ln \|x\|) = 1/x	Natural Log	⭐⭐

Trigonometric Functions

#	Formula	Name	GTU Focus
15	d/dx (sin x) = cos x	Sine	⭐⭐
16	d/dx (cos x) = −sin x	Cosine	⭐⭐
17	d/dx (tan x) = sec² x	Tangent	⭐
18	d/dx (cot x) = −csc² x	Cotangent	—
19	d/dx (sec x) = sec x · tan x	Secant	—
20	d/dx (csc x) = −csc x · cot x	Cosecant	—

Inverse Trigonometric Functions

#	Formula	Name	GTU Focus
21	d/dx (sin⁻¹ x) = 1 / √(1−x²)	Inverse Sine	⭐
22	d/dx (cos⁻¹ x) = −1 / √(1−x²)	Inverse Cosine	—
23	d/dx (tan⁻¹ x) = 1 / (x²+1)	Inverse Tangent	⭐⭐
24	d/dx (cot⁻¹ x) = −1 / (x²+1)	Inverse Cotangent	—
25	d/dx (sec⁻¹ x) = 1/(\|x\|·√(x²−1))	Inverse Secant	—
26	d/dx (csc⁻¹ x) = −1/(\|x\|·√(x²−1))	Inverse Cosecant	—

📊 Integration Formulas

Basic Rules

⭐ Most Important – Power Rule (Integration)

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)

#	Formula	Name	GTU Focus
1	∫ 1 dx = x + C	Constant	⭐⭐
2	∫ k dx = kx + C	Constant Multiple	⭐
3	∫ xⁿ dx = xⁿ⁺¹/(n+1) + C	Power Rule	⭐⭐⭐
4	∫ 1/x dx = ln \|x\| + C	Reciprocal	⭐⭐⭐
5	∫ eˣ dx = eˣ + C	Exponential (e)	⭐⭐
6	∫ aˣ dx = (1/ln a)·aˣ + C	Exponential (a)	—
7	∫ ln x dx = x·ln x − x + C	Natural Log	—

Standard Forms (Must Remember!)

#	Formula	Name	GTU Focus
8	∫ dx / √(1−x²) = sin⁻¹ x + C	Arcsin Form	⭐
9	∫ dx / (1+x²) = tan⁻¹ x + C	Arctan Form	⭐⭐
10	∫ dx / (x·√(x²−1)) = sec⁻¹ \|x\| + C	Arcsec Form	—

Trigonometric Functions

#	Formula	Name	GTU Focus
11	∫ sin x dx = −cos x + C	Sine	⭐⭐
12	∫ cos x dx = sin x + C	Cosine	⭐⭐
13	∫ tan x dx = −ln \|cos x\| + C	Tangent	⭐
14	∫ cot x dx = ln \|sin x\| + C	Cotangent	—
15	∫ sec x dx = ln \|sec x + tan x\| + C	Secant	—
16	∫ csc x dx = −ln \|csc x + cot x\| + C	Cosecant	—
17	∫ sec² x dx = tan x + C	Secant²	⭐
18	∫ csc² x dx = −cot x + C	Cosecant²	—
19	∫ sec x·tan x dx = sec x + C	Sec·Tan	—
20	∫ csc x·cot x dx = −csc x + C	Csc·Cot	—

Generalized Inverse Trigonometric Forms

#	Formula	Name	GTU Focus
21	∫ dx / √(a²−x²) = sin⁻¹(x/a) + C	Arcsin (general)	⭐
22	∫ dx / (a²+x²) = (1/a)·tan⁻¹(x/a) + C	Arctan (general)	⭐⭐
23	∫ dx / (x·√(x²−a²)) = (1/a)·sec⁻¹(\|x\|/a) + C	Arcsec (general)	—

⚠️ Common Mistakes (Avoid These!)

#	❌ Mistake	✅ Correct Way	Why it happens
1	Forgetting + C	∫ f(x) dx = F(x) + C	Indefinite integrals represent a family of functions
2	∫ sin x dx = +cos x	∫ sin x dx = −cos x + C	Derivative of cos is -sin, so integral of sin is -cos
3	d/dx (cos x) = sin x	d/dx (cos x) = −sin x	Wrong minus sign
4	Chain Rule: Forgetting inner derivative	d/dx [sin(2x)] = cos(2x)·2	Must multiply by derivative of (2x)
5	Product Rule: Using sum instead	d(uv) = u·dv + v·du	Both terms are added, not multiplied
6	∫ 1/x dx = ln(x)	∫ 1/x dx = ln \|x\| + C	Domain of ln requires absolute value for negative x
7	d/dx (xⁿ) = xⁿ⁻¹	d/dx (xⁿ) = n·xⁿ⁻¹	Forgot to multiply by the power 'n'

🎯 Exam Tips (GTU Focus)

🧠 Power Rule is your best friend

60% of problems start with d/dx (xⁿ). Master this first!

🔗 Chain Rule – always check

If you see a function inside a function (e.g., sin(3x), ln(x²+1)), apply Chain Rule. Don't forget the inner derivative!

📝 Practice PYQs

GTU frequently repeats: Chain Rule, Product Rule, Quotient Rule, ∫ 1/x dx, and ∫ eˣ dx. Solve at least 5 years of papers.

⏰ Daily Revision

Spend 10 minutes every morning writing 5–10 formulas from this sheet. Builds muscle memory!

🧪 Check your answer

After integrating, differentiate your answer. If you get the original function back → it's correct!

⏱️ Quick Revision Box (10 Second Review)

📌 Top 5 Differentiation

d/dx (xⁿ) = n·xⁿ⁻¹
d/dx (eˣ) = eˣ
d/dx (ln x) = 1/x

d/dx (sin x) = cos x
d/dx (cos x) = −sin x

📌 Top 5 Integration

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
∫ 1/x dx = ln |x| + C
∫ eˣ dx = eˣ + C

∫ sin x dx = −cos x + C
∫ cos x dx = sin x + C

⭐ Memory Trick: “sine goes to cosine (derivative), but cosine goes to minus sine.”
“Integral of sine is minus cosine, integral of cosine is sine.”

❓ Frequently Asked Questions

1. What is differentiation?

It measures the rate at which a function changes. In simple terms, it's the slope of a curve at any point.

2. What is integration?

It's the reverse process of differentiation. It helps find areas under curves, total accumulated values, and is used heavily in physics and engineering.

3. Why do we add + C in integration?

Because the derivative of any constant is zero. So, when we integrate, there might be an unknown constant hidden in the original function. We use C to represent that.

4. Which formulas are most important for GTU?

Must-know: Power Rule, Chain Rule, Product Rule, ∫ 1/x dx, ∫ eˣ dx, ∫ sin x dx, ∫ cos x dx, and tan⁻¹ x forms.

5. How to remember trig derivatives?

“Sine is nice, Cosine is negative sine.” For tangent, remember sec². For cotangent, remember −csc².

6. What is the difference between derivative and integral?

Derivative = Rate of change. Integral = Accumulation / Area under the curve. They are inverse operations (Fundamental Theorem of Calculus).