All Differentiation & Integration Rules for NEET — Complete Reference

🧒 0 · Explained for a 10-year-old (with animation)

Before any formulas, here's what differentiation and integration really are — using two simple animated pictures.

0.1 Differentiation = "How fast is it changing right now?"

Think of riding a bike on a hilly road. Your speedometer doesn't show your total trip — it shows how fast you are going right at this moment. That's exactly what differentiation does.

You give it a curve (your height above the ground, your bank balance over time, the position of a ball), and at any point along that curve it tells you "how steeply is it changing right now?"

Going UP a hill

Slope is positive. The bigger the climb, the bigger the number. dy/dx > 0

At the TOP of a hill

Slope is zero — for a tiny moment you're flat. dy/dx = 0 (this is how we find peaks and valleys!)

Going DOWN a hill

Slope is negative. dy/dx < 0

Flat road

Slope is zero everywhere — nothing is changing. dy/dx = 0

🎈 In one sentence

Differentiation answers "how fast is something changing at this exact moment?" In physics that becomes velocity (how fast position changes) and acceleration (how fast velocity changes).

0.2 Integration = "How much altogether?"

Think of filling a swimming pool with a hose. The hose might run fast at first, slow down for a while, then speed up again. To know the total water in the pool after some time, you can't just look at the speed at one moment — you have to add up everything that has flowed.

Integration is the maths version of that adding-up. Take the curve (the flow rate, the height of a hill, anything), slice it into super-thin vertical strips, multiply each strip's height by its tiny width dx, and sum them all. The result is the total area under the curve — and it tells you how much water reached the pool, how far the car drove, or how much charge flowed through a wire.

Tall slices

Curve is high → each rectangle is tall → that section contributes a LOT to the total.

Short slices

Curve is low → small contribution. The "small bit" still counts!

Curve below zero

Slice points DOWN — counts as NEGATIVE area (water leaves the pool).

Thinner slices

The thinner you make them, the more accurate your total. In the limit (dx → 0) the answer is EXACT.

🎈 In one sentence

Integration answers "how much, in total, has happened?" In physics that becomes distance (total of velocity over time), charge (total of current over time), or work (total of force over distance).

0.3 The two are opposites — the magic link

If differentiation breaks something into "how fast" pieces, then integration glues them back together. Apply one and then the other, you get back what you started with — they are perfect opposites.

That is why every formula in this page comes in pairs. Every derivative on the left side has a matching integral on the right. Master the derivatives and the integrals become much easier — they are just derivatives read backwards.

∂ 1 · What is Differentiation?

Differentiation finds the instantaneous rate of change of a quantity — the slope of a curve at one point.

dy/dx = lim_{Δx → 0} [y(x + Δx) − y(x)] / Δx

🎈 Plain English

Take two points on the curve very close together. Compute rise/run. Shrink the gap to zero. That number is the slope at the chosen point — its derivative.

1.1 Notation — three ways to write the same thing

Notation	Read as	Used by
`dy/dx`	"derivative of y with respect to x"	Leibniz (standard NEET)
`f′(x)`	"f-prime of x"	Lagrange (most textbooks)
`ẏ or ẋ`	"derivative w.r.t. time"	Newton (physics)

1.2 The slope question for common curves

Curve	Slope at any x	Slope at x = 1
`y = x`	1	1
`y = x²`	2x	2
`y = x³`	3x²	3
`y = sin x`	cos x	cos 1 ≈ 0.54
`y = eˣ`	eˣ	e ≈ 2.72

∂ 2 · The Five Rules of Differentiation

Rule	Formula	Plain meaning
Constant	`d/dx (c) = 0`	Constants don't change → slope is 0.
Constant multiple	`d/dx (c · f) = c · f′`	Constants slide outside the derivative.
Sum / Difference	`d/dx (u ± v) = u′ ± v′`	Differentiate each piece, add/subtract.
Power	`d/dx (xⁿ) = n · xⁿ⁻¹`	Drop the power down, subtract 1.
Product	`d/dx (u·v) = u′·v + u·v′`	"First × derivative of second + second × derivative of first."
Quotient	`d/dx (u/v) = (u′·v − u·v′) / v²`	Mind the minus sign and the v² in the denominator.
Chain	`d/dx (f(g(x))) = f′(g(x)) · g′(x)`	Differentiate outer, then multiply by derivative of inner.

🎈 Power rule memory trick

For xⁿ: "drop the power to the front, then drop the power by one." So x³ → 3x² · x → 1 · x⁵ → 5x⁴.

2.1 Tiny worked steps

📌 d/dx of 5x³ + 2x − 7

5·3·x² + 2 − 0 = 15x² + 2

📌 d/dx of (x² + 1)(x − 3) — product rule

u = x²+1, u′ = 2x. v = x−3, v′ = 1.
Result = 2x(x−3) + (x²+1)(1) = 2x²−6x + x²+1 = 3x² − 6x + 1

📌 d/dx of sin(3x²) — chain rule

Outer = sin(·), derivative = cos(·). Inner = 3x², derivative = 6x.
Result = cos(3x²) · 6x = 6x cos(3x²)

∂ 3 · Standard Derivatives — the must-know table

f(x)	f′(x)	f(x)	f′(x)
`c` (constant)	0	`xⁿ`	`n·xⁿ⁻¹`
`x`	1	`1/x`	`−1/x²`
`√x`	`1/(2√x)`	`∛x`	`1/(3 x²^⁄³)`
`eˣ`	`eˣ`	`aˣ`	`aˣ · ln a`
`ln x`	`1/x`	`log_a x`	`1/(x · ln a)`
`\|x\|`	`x/\|x\|` (sign)	`eᵏˣ`	`k · eᵏˣ`

∂ 4 · Trigonometric & Inverse Trig Derivatives

4.1 Trigonometric functions

f(x)	f′(x)	f(x)	f′(x)
`sin x`	`cos x`	`cos x`	`−sin x`
`tan x`	`sec²x`	`cot x`	`−cosec²x`
`sec x`	`sec x · tan x`	`cosec x`	`−cosec x · cot x`

4.2 Inverse trig functions

f(x)	f′(x)	Domain
`sin⁻¹ x`	`1/√(1 − x²)`	−1 < x < 1
`cos⁻¹ x`	`−1/√(1 − x²)`	−1 < x < 1
`tan⁻¹ x`	`1/(1 + x²)`	all x
`cot⁻¹ x`	`−1/(1 + x²)`	all x
`sec⁻¹ x`	`1/(\|x\|·√(x² − 1))`	\|x\| > 1
`cosec⁻¹ x`	`−1/(\|x\|·√(x² − 1))`	\|x\| > 1

🎈 Memory pattern

Every inverse-trig pair (sin⁻¹ / cos⁻¹, tan⁻¹ / cot⁻¹, sec⁻¹ / cosec⁻¹) has the same derivative magnitude with opposite signs.

∂ 5 · Exponential, Logarithmic & Hyperbolic

f(x)	f′(x)
`eˣ`	`eˣ` (its own derivative)
`e^kx`	`k · e^kx`
`e^f(x)`	`e^f(x) · f′(x)`
`aˣ`	`aˣ · ln a`
`ln x`	`1/x`
`ln(f(x))`	`f′(x)/f(x)`
`log_a x`	`1/(x · ln a)`
`sinh x`	`cosh x`
`cosh x`	`sinh x`
`tanh x`	`sech²x`

∂ 6 · Implicit & Logarithmic Differentiation

6.1 Implicit differentiation

Used when y is defined implicitly by an equation in x and y, not as y = f(x). Differentiate every term w.r.t. x, treating y as a function of x; collect dy/dx.

📌 Example

x² + y² = 25. Differentiate: 2x + 2y·(dy/dx) = 0 → dy/dx = −x/y.

6.2 Logarithmic differentiation — for y = (something)^something

When the function has the variable in BOTH base and exponent (e.g. x^x, (sin x)^{cos x}), use:

Take ln of both sides.
Use log laws to split into a sum.
Differentiate both sides w.r.t. x.
Multiply through by y.

📌 Example — y = xˣ

ln y = x · ln x.
(1/y) dy/dx = ln x + x · (1/x) = ln x + 1.
So dy/dx = xˣ (ln x + 1).

6.3 Parametric differentiation

If x = x(t) and y = y(t):

dy/dx = (dy/dt) / (dx/dt)

📌 Example — x = a cos θ, y = a sin θ

dx/dθ = −a sin θ. dy/dθ = a cos θ.
dy/dx = (a cos θ)/(−a sin θ) = −cot θ

∂ 7 · Higher-Order Derivatives

Order	Symbol	Physics meaning
1st	`dy/dx`, `f′(x)`	Velocity (when y is position, x is t)
2nd	`d²y/dx²`, `f″(x)`	Acceleration
3rd	`d³y/dx³`, `f‴(x)`	Jerk (rarely tested)
nth	`dⁿy/dxⁿ`, `f⁽ⁿ⁾(x)`	General

📌 Example — SHM

If x(t) = A sin(ωt), then v = dx/dt = Aω cos(ωt), a = d²x/dt² = −Aω² sin(ωt) = −ω²x.
The minus sign in a = −ω²x is the signature of SHM.

∂ 8 · Where Differentiation Appears in NEET

8.1 Physics — Kinematics

v = dx/dt a = dv/dt = d²x/dt²

8.2 Physics — Electromagnetism

EMF: ε = − dΦ/dt (Faraday's law)

I = dQ/dt V_L = L · dI/dt

8.3 Physics — Maxima & Minima

To find a maximum or minimum of f(x):

Solve f′(x) = 0 for stationary points.
If f″(x) < 0 at that point → maximum (curve bends down).
If f″(x) > 0 at that point → minimum (curve bends up).

⚛️ NEET classic

Projectile range R = (u²/g)·sin(2θ). Set dR/dθ = 0: cos(2θ) = 0 → θ = 45°. That's why 45° gives the maximum range.

8.4 Chemistry — Kinetics

Rate = − d[reactant]/dt = + d[product]/dt

8.5 Chemistry — Thermodynamics partials

(∂G/∂T)_p = − S (∂G/∂p)_T = V

— End of Differentiation · Start of Integration —

∫ 9 · What is Integration?

Integration is the reverse of differentiation — it finds the function whose derivative is the given one. Geometrically, the definite integral of f(x) from a to b equals the area under the curve y = f(x) between x = a and x = b.

If F′(x) = f(x), then ∫ f(x) dx = F(x) + C

🎈 Plain English

If differentiation says "find the slope," integration says "given the slope at every point, find back the curve." The +C is the constant of integration — different curves can have the same slope (parallel curves shifted up or down).

9.1 Two basic properties

Property	Formula
Constant multiple	`∫ c·f(x) dx = c · ∫ f(x) dx`
Sum / Difference	`∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx`

∫ 10 · Standard Integral Table

10.1 Power, exponential, log

f(x)	∫ f(x) dx
`xⁿ` (n ≠ −1)	`xⁿ⁺¹ / (n + 1) + C`
`1/x`	`ln\|x\| + C`
`eˣ`	`eˣ + C`
`e^kx`	`e^kx/k + C`
`aˣ`	`aˣ / ln a + C`
`1/(x + a)`	`ln\|x + a\| + C`

10.2 Trigonometric integrals

f(x)	∫ f(x) dx
`sin x`	`−cos x + C`
`cos x`	`sin x + C`
`tan x`	`− ln\|cos x\| + C or ln\|sec x\| + C`
`cot x`	`ln\|sin x\| + C`
`sec x`	`ln\|sec x + tan x\| + C`
`cosec x`	`ln\|cosec x − cot x\| + C`
`sec²x`	`tan x + C`
`cosec²x`	`−cot x + C`
`sec x · tan x`	`sec x + C`
`cosec x · cot x`	`−cosec x + C`

10.3 Inverse trig integrals

f(x)	∫ f(x) dx
`1/√(1 − x²)`	`sin⁻¹ x + C`
`−1/√(1 − x²)`	`cos⁻¹ x + C`
`1/(1 + x²)`	`tan⁻¹ x + C`
`1/(\|x\|·√(x² − 1))`	`sec⁻¹ x + C`

⚡ Integration table = derivatives table read backwards. If you know derivatives cold, you know two-thirds of integrals automatically.

∫ 11 · Integration by Substitution

If the integrand contains a function and its derivative (or a near-multiple), substitute u = inner function.

∫ f(g(x)) · g′(x) dx = ∫ f(u) du (where u = g(x))

11.1 Common patterns

Integrand	Substitute	Result
`∫ (f(x))ⁿ · f′(x) dx`	u = f(x)	`(f(x))ⁿ⁺¹ / (n+1) + C`
`∫ f′(x)/f(x) dx`	u = f(x)	`ln\|f(x)\| + C`
`∫ f′(x) · e^f(x) dx`	u = f(x)	`e^f(x) + C`
`∫ sin(ax + b) dx`	u = ax + b	`−cos(ax+b)/a + C`
`∫ (ax + b)ⁿ dx`	u = ax + b	`(ax+b)ⁿ⁺¹ / [a(n+1)] + C`

📌 Example

∫ 2x / (x² + 1) dx. Let u = x² + 1, du = 2x dx. Integral = ∫ du/u = ln|u| + C = ln(x²+1) + C.

∫ 12 · Integration by Parts

Used when the integrand is a product of two functions that don't share a clean substitution.

∫ u dv = u·v − ∫ v du

12.1 The "ILATE / LIATE" rule for picking u

Pick u from the function class that comes first in this priority list:

I — Inverse trig (sin⁻¹ x, tan⁻¹ x …)
L — Logarithmic (ln x, log x)
A — Algebraic (xⁿ, polynomials)
T — Trigonometric (sin x, cos x)
E — Exponential (eˣ, aˣ)

📌 Example — ∫ x · eˣ dx

u = x (algebraic), dv = eˣ dx. du = dx, v = eˣ.
∫ x eˣ dx = x · eˣ − ∫ eˣ dx = x eˣ − eˣ + C = eˣ (x − 1) + C

📌 Example — ∫ ln x dx

u = ln x, dv = dx. du = dx/x, v = x.
∫ ln x dx = x ln x − ∫ x · (1/x) dx = x ln x − x + C

12.2 Standard "by parts" result

∫ eˣ [f(x) + f′(x)] dx = eˣ · f(x) + C

∫ 13 · Special / Standard Integrals

These should be memorised cold — they appear repeatedly in NEET physics problems.

13.1 Quadratic-denominator integrals

Integrand	Answer
`1/(a² + x²)`	`(1/a) · tan⁻¹(x/a) + C`
`1/(a² − x²)`	`(1/2a) · ln\|(a + x)/(a − x)\| + C`
`1/(x² − a²)`	`(1/2a) · ln\|(x − a)/(x + a)\| + C`
`1/√(a² − x²)`	`sin⁻¹(x/a) + C`
`1/√(a² + x²)`	`ln\|x + √(a² + x²)\| + C`
`1/√(x² − a²)`	`ln\|x + √(x² − a²)\| + C`

13.2 Square-root integrals

Integrand	Answer
`√(a² − x²)`	`(x/2)√(a²−x²) + (a²/2) sin⁻¹(x/a) + C`
`√(a² + x²)`	`(x/2)√(a²+x²) + (a²/2) ln\|x + √(a²+x²)\| + C`
`√(x² − a²)`	`(x/2)√(x²−a²) − (a²/2) ln\|x + √(x²−a²)\| + C`

13.3 Reduction-style

Integrand	Answer
`eᵃˣ · sin(bx)`	`e^ax(a sin bx − b cos bx)/(a² + b²) + C`
`eᵃˣ · cos(bx)`	`e^ax(a cos bx + b sin bx)/(a² + b²) + C`

∫ 14 · Integration by Partial Fractions

For rational functions P(x)/Q(x) where deg(P) < deg(Q), split Q into simpler factors and write the rational as a sum.

Form of Q(x)	Partial fraction
(x − a)(x − b)	`A/(x−a) + B/(x−b)`
(x − a)²	`A/(x−a) + B/(x−a)²`
(x − a)(x² + bx + c)	`A/(x−a) + (Bx + C)/(x² + bx + c)`
(x² + bx + c)²	`(A x + B)/(x² + bx + c) + (Cx + D)/(x² + bx + c)²`

📌 Example

∫ dx/(x² − 1) = ∫ dx / [(x−1)(x+1)] = ½ ∫ [1/(x−1) − 1/(x+1)] dx = ½ ln|(x−1)/(x+1)| + C

∫ 15 · Definite Integrals & Their Properties

∫_a^b f(x) dx = F(b) − F(a) (where F′ = f)

This is the Fundamental Theorem of Calculus — it links differentiation and integration.

15.1 Properties of definite integrals

Property	Formula
Reverse limits	`∫_a^b f(x) dx = − ∫_b^a f(x) dx`
Same limits	`∫_a^a f(x) dx = 0`
Split point	`∫_a^b f(x) dx = ∫_a^c f dx + ∫_c^b f dx`
King property	`∫_a^b f(x) dx = ∫_a^b f(a + b − x) dx`
Even integrand on (−a, a)	`= 2 · ∫₀^a f(x) dx` if f(−x) = f(x)
Odd integrand on (−a, a)	`= 0` if f(−x) = −f(x)
Periodic (period T)	`∫₀^nT f(x) dx = n · ∫₀^T f(x) dx`

15.2 Area under a curve

Area under y = f(x) between x = a and x = b is ∫_a^b f(x) dx (if f ≥ 0)

Area between two curves: ∫_a^b [f_upper(x) − f_lower(x)] dx

∫ 16 · Where Integration Appears in NEET

16.1 Physics — Kinematics

v(t) = v₀ + ∫₀^t a(t) dt x(t) = x₀ + ∫₀^t v(t) dt

⚛️ Plain English

If you know acceleration as a function of time, you INTEGRATE to find velocity. Integrate velocity to find position. Differentiation goes the other way (position → velocity → acceleration).

16.2 Physics — Work, Energy & Power

W = ∫ F · dx (work done by a variable force)

U(x) = −∫ F(x) dx + C (potential energy)

16.3 Physics — Electromagnetism

Q = ∫ I dt (charge flowing in time interval)

V = ∫ E · dl (potential difference)

Φ = ∫ B · dA (magnetic flux)

16.4 Chemistry — Integrated rate laws

Order	Differential form	Integrated form
Zero	`−d[A]/dt = k`	`[A] = [A]₀ − kt`
First	`−d[A]/dt = k[A]`	`ln[A] = ln[A]₀ − kt`
Second	`−d[A]/dt = k[A]²`	`1/[A] = 1/[A]₀ + kt`

16.5 Chemistry — Thermodynamic work

W_{reversible isothermal} = − nRT · ln(V₂/V₁) = − ∫_V₁^V₂ p dV

🔑 17 · One-Page Cheat Sheet

Differentiation

f(x)	f′(x)
`c`	0
`xⁿ`	`n xⁿ⁻¹`
`eˣ`	`eˣ`
`aˣ`	`aˣ ln a`
`ln x`	`1/x`
`sin x / cos x`	`cos x / −sin x`
`tan x / cot x`	`sec²x / −cosec²x`
`sec x / cosec x`	`sec x tan x / −cosec x cot x`
`sin⁻¹ x / cos⁻¹ x`	`±1/√(1−x²)`
`tan⁻¹ x`	`1/(1+x²)`
Product	`u′v + uv′`
Quotient	`(u′v − uv′)/v²`
Chain	`f′(g(x))·g′(x)`
Max / min	f′ = 0; check sign of f″

Integration

f(x)	∫ f(x) dx
`xⁿ` (n ≠ −1)	`xⁿ⁺¹/(n+1) + C`
`1/x`	`ln\|x\| + C`
`eˣ`	`eˣ + C`
`aˣ`	`aˣ/ln a + C`
`sin x`	`−cos x + C`
`cos x`	`sin x + C`
`sec²x`	`tan x + C`
`1/(1+x²)`	`tan⁻¹ x + C`
`1/√(1−x²)`	`sin⁻¹ x + C`
`1/(a²+x²)`	`(1/a) tan⁻¹(x/a) + C`
`1/√(a²−x²)`	`sin⁻¹(x/a) + C`
By parts	`∫ u dv = uv − ∫ v du` (ILATE)
Substitution	u = inner function; replace dx
Definite	`∫_a^b f dx = F(b) − F(a)`

⚡ Memory anchors
1) Power → drop, subtract. Inverse: Power → add, raise.
2) Three product/quotient/chain rules + ILATE solve 90 % of NEET integrals.
3) The constants +C, ln a, and 1/(n+1) are the three things students forget most — flag them.

🧒 0 · Explained for a 10-year-old (with animation)

0.1 Differentiation = "How fast is it changing right now?"

Going UP a hill

At the TOP of a hill

Going DOWN a hill

Flat road

0.2 Integration = "How much altogether?"

Tall slices

Short slices

Curve below zero

Thinner slices

0.3 The two are opposites — the magic link

∂ 1 · What is Differentiation?

1.1 Notation — three ways to write the same thing

1.2 The slope question for common curves

∂ 2 · The Five Rules of Differentiation

2.1 Tiny worked steps

∂ 3 · Standard Derivatives — the must-know table

∂ 4 · Trigonometric & Inverse Trig Derivatives

4.1 Trigonometric functions

4.2 Inverse trig functions

∂ 5 · Exponential, Logarithmic & Hyperbolic

∂ 6 · Implicit & Logarithmic Differentiation

6.1 Implicit differentiation

6.2 Logarithmic differentiation — for y = (something)something

6.3 Parametric differentiation

∂ 7 · Higher-Order Derivatives

∂ 8 · Where Differentiation Appears in NEET

8.1 Physics — Kinematics

8.2 Physics — Electromagnetism

8.3 Physics — Maxima & Minima

8.4 Chemistry — Kinetics

8.5 Chemistry — Thermodynamics partials

∫ 9 · What is Integration?

9.1 Two basic properties

∫ 10 · Standard Integral Table

10.1 Power, exponential, log

10.2 Trigonometric integrals

10.3 Inverse trig integrals

∫ 11 · Integration by Substitution

11.1 Common patterns

∫ 12 · Integration by Parts

12.1 The "ILATE / LIATE" rule for picking u

12.2 Standard "by parts" result

∫ 13 · Special / Standard Integrals

13.1 Quadratic-denominator integrals

13.2 Square-root integrals

13.3 Reduction-style

∫ 14 · Integration by Partial Fractions

∫ 15 · Definite Integrals & Their Properties

15.1 Properties of definite integrals

15.2 Area under a curve

∫ 16 · Where Integration Appears in NEET

16.1 Physics — Kinematics

16.2 Physics — Work, Energy & Power

16.3 Physics — Electromagnetism

16.4 Chemistry — Integrated rate laws

16.5 Chemistry — Thermodynamic work

🔑 17 · One-Page Cheat Sheet

Differentiation

Integration

6.2 Logarithmic differentiation — for y = (something)^something