Limits, Continuity and Differentiability – JEE Main Maths Formula Sheet & Class 12 Notes | All Formulas, Results & Differentiation Rules

Definition of Limit, Existence, and Indeterminate Forms

Why LHL = RHL and Indeterminate Form Classification Are Core JEE Main Topics

Definition of Limit (Limits, Continuity and Differentiability — from PDF): A real number l is called the limit of f(x) as x → a, written lim_{x→a} f(x) = l, if for every ε > 0 there exists δ = δ(ε) > 0 such that |f(x) – l| < ε whenever 0 < |x – a| < δ. The function value f(a) is irrelevant to the limit — only the behaviour as x approaches a matters.

Existence of Limit (from PDF — Limits Continuity and Differentiability): lim_{x→a} f(x) exists if and only if LHL = RHL = finite and definite:

LHL = lim_{x→a⁻} f(x) = lim_{h→0} f(a–h)

RHL = lim_{x→a⁺} f(x) = lim_{h→0} f(a+h)

The limit lim_{x→a} f(x) exists iff LHL = RHL (both finite and equal).

Indeterminate forms in Limits, Continuity and Differentiability (from PDF — JEE Main): A form whose value cannot be uniquely determined. The 7 standard indeterminate forms are:

0/0, ∞/∞, 0×∞, ∞–∞, 1^∞, 0⁰, ∞⁰

Explanation of 0/0 from PDF: if k = 0/0, then k×0 = 0 for infinitely many values of k, so k is not fixed. Every one of the 7 forms represents a situation where the limit may exist but cannot be read directly from the expression. Each requires a specific technique to resolve.

Resolving indeterminate forms in Limits, Continuity and Differentiability (JEE Main): 0/0 and ∞/∞ → use L'Hospital's rule OR factorisation OR expansion. 0×∞ and ∞–∞ → rewrite as 0/0 or ∞/∞ using algebra (LCM, rationalisation). 1^∞, 0⁰, ∞⁰ → take logarithm to reduce to 0×∞ then to 0/0 or ∞/∞. Download the Free PDF for Limits, Continuity and Differentiability for all indeterminate form worked examples for JEE Main.

Algebra of Limits and Sandwich Theorem — Limits, Continuity and Differentiability

Why the 9 Algebra of Limits Rules Are the Computational Foundation for JEE Main

If lim_{x→a} f(x) = L and lim_{x→a} g(x) = M (both exist finitely), then from the PDF — Limits, Continuity and Differentiability:

9 Algebra of Limits rules (from PDF — JEE Main):

(i) lim[f(x) ± g(x)] = lim f(x) ± lim g(x) = L ± M

(ii) lim[f(x)/g(x)] = lim f(x)/lim g(x) = L/M, provided M ≠ 0

(iii) lim[k·f(x)] = k·lim f(x) = kL (k is a constant)

(iv) lim[f(x)]^g(x) = [lim f(x)]^[lim g(x)] = L^M

(v) lim e^f(x) = e^(lim f(x)) = e^L

(vi) lim log f(x) = log(lim f(x)) = log L (provided L > 0)

(vii) lim[f(x)·g(x)] = [lim f(x)]·[lim g(x)] = L·M

(viii) lim (fog)(x) = f(lim g(x)) = f(M), provided f is continuous at M (from PDF)

(ix) Sandwich Theorem (Squeeze Theorem): If f(x) ≤ h(x) ≤ g(x) for all x in a neighbourhood of a, and lim_{x→a} f(x) = lim_{x→a} g(x) = L, then lim_{x→a} h(x) = L. The squeeze theorem is used when the limit of h(x) cannot be found directly but can be bounded between two functions whose limits are known.

L'Hospital's Rule (from PDF — Limits, Continuity and Differentiability): If f(a) = 0 and g(a) = 0 (or both → ∞), then lim_{x→a} f(x)/g(x) = lim_{x→a} f'(x)/g'(x). Differentiate numerator and denominator separately (NOT using quotient rule). Apply repeatedly until the indeterminate form is removed. Works for both 0/0 and ∞/∞ forms. Also applies to x → ∞ cases. Note from PDF: for 0×∞ and ∞–∞ forms, first convert to 0/0 or ∞/∞ before applying L'Hospital's. For 1^∞, 0⁰, ∞⁰: take log, reduce to 0×∞ form, then convert. Download the Free PDF for Limits, Continuity and Differentiability for L'Hospital's rule examples.

All 15 Standard Limits — Direct from the PDF for JEE Main

Why Standard Limits Are the Most Directly Tested Limits, Continuity and Differentiability Formulas

The following standard limits are listed directly from the Aakash PDF — every one appears in JEE Main either as a direct question or as a step in multi-part limit evaluation.

Trigonometric standard limits in Limits, Continuity and Differentiability (from PDF):

1. lim_{x→0} (sin x)/x = 1; lim_{x→0} (tan x)/x = 1 [equivalent to lim (sin x/x) = lim (tan x/x) = 1 as x→0]

2. lim_{x→0} cos x = 1 and lim_{x→0} sec x = 1

3. lim_{x→0} (sin⁻¹x)/x = 1 and lim_{x→0} (tan⁻¹x)/x = 1

4. lim_{x→∞} (sin x)/x = 0 [x in radians, sin x bounded, 1/x→0]

5. lim_{x→∞} x·sin(1/x) = 1 [since t = 1/x → 0, expression = sint/t → 1]

6. lim_{x→0} [sin x]/x = 0 [where [ ] = greatest integer function; sin x < x for x > 0, so [sinx/x] = 0 for small x⁺]

7. lim_{x→0} [tan x]/x = 1 [note: tan x > x for 0 < x < π/2, so [tanx/x] = 1]

8. lim_{x→0} cos(x) = 0 [actually lim_{x→0} (cos x)^(1/x) ... from PDF: lim_{x→0} x·cos(x) = 0]

9. lim_{x→0} [tan⁻¹x]/x = 1 [greatest integer of tan⁻¹x/x → ⌊1⌋ = 1]

Note from PDF: sin x < x < tan x for 0 < x < π/2.

Algebraic and exponential standard limits (from PDF — Limits, Continuity and Differentiability JEE Main):

10. lim_{x→a} (xⁿ – aⁿ)/(x – a) = naⁿ⁻¹ for a > 0, n ∈ ℝ; and if a < 0, n ∈ ℤ

11. lim_{x→a} (xⁿ – aⁿ)/(xᵐ – aᵐ) = (n/m)·a^(n–m) [from PDF, follows from formula 10]

12. lim_{x→0} (aˣ – 1)/x = log_e a for a > 0; equivalently lim_{x→0} (eˣ – 1)/x = 1

13. lim_{x→0} log_e(1+x)/x = 1 (= lim_{x→0} ln(1+x)/x) and lim_{x→0} log_a(1+x)/x = log_a e

14. lim_{n→∞} (1 + 1/n)ⁿ = e; lim_{x→0} (1 + x)^(1/x) = e (two equivalent forms of the definition of e)

15. lim_{x→0} (1 + λx)^(1/x) = e^λ for any constant λ; lim_{n→∞} (1 + λ/n)ⁿ = e^λ

Applying standard limits using substitution in Limits, Continuity and Differentiability (JEE Main technique): If lim_{x→a} f(x) = 0 then replace f(x) with u = f(x) → 0 and use: lim sin(f(x))/f(x) = 1; lim (a^f(x) – 1)/f(x) = ln a; lim log(1+f(x))/f(x) = 1. Download the Free PDF for Limits, Continuity and Differentiability for all standard limit applications for JEE Main.

Important Expansions — From PDF for Limit Evaluation and Series

Why Taylor-Maclaurin Expansions Are the Most Powerful Limits, Continuity and Differentiability Tool for JEE Main

The following expansions from the PDF are used to evaluate limits, find coefficient-of-xⁿ type problems, and connect to the Binomial Theorem and Sequences chapters in JEE Main.

Complete expansion list from PDF — Limits, Continuity and Differentiability:

1. aˣ = 1 + x(ln a) + x²(ln a)²/2! + x³(ln a)³/3! + … for a > 0 (from PDF, general exponential)

2. eˣ = 1 + x/1! + x²/2! + x³/3! + … (valid for all x ∈ ℝ)

3. ln(1+x) = x – x²/2 + x³/3 – x⁴/4 + … (–1 < x ≤ 1) [from PDF, note log = ln = log_e]

4. sin x = x – x³/3! + x⁵/5! – x⁷/7! + …

5. cos x = 1 – x²/2! + x⁴/4! – x⁶/6! + …

6. tan x = x + x³/3 + 2x⁵/15 + … (for |x| < π/2)

7. sin⁻¹x = x + (1²/3!)x³ + (1²·3²/5!)x⁵ + … = x + x³/6 + 3x⁵/40 + … (|x| ≤ 1) [from PDF]

8. tan⁻¹x = x – x³/3 + x⁵/5 – x⁷/7 + … (|x| ≤ 1)

9. (1+x)ⁿ = 1 + nx + n(n–1)x²/2! + … (generalised Binomial, |x| < 1)

Using expansions to evaluate limits (Limits, Continuity and Differentiability — JEE Main method): Substitute the expansion for each function, cancel common factors, and take the limit as x → 0. Example: lim_{x→0} (sinx – x)/x³ = lim_{x→0} [(x – x³/6 + …) – x]/x³ = lim_{x→0} (–x³/6 + …)/x³ = –1/6. This method works for any 0/0 limit involving standard functions at x → 0. The expansion method from Limits, Continuity and Differentiability is faster than L'Hospital's for many JEE Main questions. Download the Free PDF for all expansion-based limit examples.

Evaluation of 1^∞ and Exponential Limits — From PDF

Why the 1^∞ Form Is the Most-Tested Indeterminate Limit in JEE Main

The 1^∞ indeterminate form in Limits, Continuity and Differentiability arises when f(x) → 1 and g(x) → ∞ as x → a. Evaluating [f(x)]^g(x) directly gives 1^∞ which is indeterminate.

Two evaluation methods for 1^∞ form (from PDF — Limits, Continuity and Differentiability):

Method (a) — Standard 1^∞ shortcut from PDF: If lim_{x→a} f(x) = 1 and lim_{x→a} g(x) = ∞, then:

lim_{x→a} [f(x)]^g(x) = e^{lim_{x→a} g(x)·[f(x)–1]}

This is the direct 1^∞ formula. Memorise it — it makes 1^∞ problems one-line calculations in JEE Main.

More precisely from PDF: if lim_{x→a} f(x) → 1 and lim_{x→a} g(x) → ∞, then:

lim [f(x)]^g(x) = lim [1 + (f(x)–1)]^g(x) = e^{lim g(x)·(f(x)–1)} (using the standard result lim(1+t)^(1/t) = e)

Method (b) — Logarithm approach from PDF: If lim [f(x)]^g(x) gives 0^∞ or ∞^0 form: take natural log → ln[f(x)^g(x)] = g(x)·ln f(x) → becomes 0×∞ form → convert to 0/0 → apply L'Hospital's → exponentiate to get the answer.

Standard e-form limits in Limits, Continuity and Differentiability (from PDF):

lim_{n→∞} (1 + 1/n)ⁿ = e

lim_{x→0} (1 + x)^(1/x) = e

lim_{x→0} (1 + λx)^(1/x) = e^λ

lim_{x→∞} (1 + λ/x)^x = e^λ

lim_{x→0} [(1+x)^(1/x) – e]/x = –e/2 (2nd order refinement)

Newton-Leibnitz Theorem for limits (from PDF — Limits, Continuity and Differentiability): If k(x) = ∫_{α(x)}^{β(x)} f(t) dt, then k'(x) = f(β(x))·β'(x) – f(α(x))·α'(x). This is used to evaluate limits of the type lim_{x→a} [∫_{α(x)}^{β(x)} f(t)dt] / g(x) where numerator → 0 and denominator → 0. Differentiate numerator (by Newton-Leibnitz) and denominator (by chain rule) and take the limit. Download the Free PDF for Limits, Continuity and Differentiability for 1^∞ form and Newton-Leibnitz examples for JEE Main.

Continuity — Definition, Types of Discontinuity, and All Properties

Why Continuity Analysis Is the Most Concept-Heavy Limits, Continuity and Differentiability JEE Main Section

Continuity at a point (from PDF — Limits, Continuity and Differentiability): f(x) is continuous at x = a if: f(a⁻) = f(a⁺) = f(a) = a finite number. Equivalently: lim_{x→a} f(x) exists finitely, f(a) is defined, and lim_{x→a} f(x) = f(a). Formal ε-δ definition from PDF: f is continuous at a if for every ε > 0, there exists δ > 0 such that 0 ≤ |x–a| < δ ⟹ |f(x)–f(a)| < ε.

Continuity in open interval (a, b): f is continuous in (a, b) if it is continuous at every point of (a, b).

Continuity in closed interval [a, b] (from PDF — 3 conditions): (1) f is continuous from the right at x = a: lim_{h→0} f(a+h) = f(a). (2) f is continuous from the left at x = b: lim_{h→0} f(b–h) = f(b). (3) f is continuous at every point of the open interval (a, b).

10 Properties of Continuous Functions (from PDF — Limits, Continuity and Differentiability):

1. If f and g are continuous at x = a, then: f+g, f–g, fg, f/g (if g(a)≠0), kf are all continuous at a.

2. [f(x)]^(m/n) is continuous at x = a, provided it is defined in an interval containing a.

3. If f is continuous at a and g is continuous at f(a), then gof is continuous at a (composition of continuous functions is continuous).

4. If f is continuous at a and g is discontinuous at a, then f+g and f–g are discontinuous at a, but fg may still be continuous at a.

5. If f is continuous at a and f(a) ≠ 0, then there exists an open interval (a–δ, a+δ) such that f(x) has the same sign as f(a) throughout this interval.

6. Intermediate Value Theorem (IVT): If f is continuous on [a, b] and f(a)·f(b) < 0, then there exists at least one c ∈ (a, b) such that f(c) = 0 (root exists between a and b).

7. If f is continuous on [a, b] and k is any real number between f(a) and f(b), then there exists at least one c ∈ (a, b) such that f(c) = k (General IVT).

8. If f is continuous on [a, b], then f is bounded on [a, b] (there exists M such that |f(x)| ≤ M for all x ∈ [a, b]).

9. Every polynomial is continuous at every point of ℝ.

10. Every rational function p(x)/q(x) is continuous at every point where q(x) ≠ 0. Logarithmic, exponential, trigonometric, inverse circular, and absolute value functions are all continuous in their domains.

Types of Discontinuity (from PDF — Limits, Continuity and Differentiability):

Removable discontinuity: LHL = RHL = finite value ≠ f(a) (limit exists but ≠ function value). Can be "removed" by redefining f(a).

Discontinuity of 1st kind (jump discontinuity): LHL ≠ RHL but both exist (both are finite). Cannot be removed. Jump = |RHL – LHL|.

Discontinuity of 2nd kind (infinite discontinuity): At least one of LHL or RHL does not exist or is ±∞. Download the Free PDF for Limits, Continuity and Differentiability for all continuity and discontinuity analysis examples for JEE Main.

Differentiability — LHD, RHD, Relation to Continuity

Why Differentiability Analysis Is a Direct JEE Main Limits, Continuity and Differentiability Question

Differentiability at a point (from PDF — Limits, Continuity and Differentiability): f(x) is differentiable at x = a if the derivative exists at x = a, which requires:

Left Hand Derivative (LHD): f'(a⁻) = lim_{h→0} [f(a–h) – f(a)] / (–h) = lim_{h→0⁺} [f(a) – f(a–h)] / h

Right Hand Derivative (RHD): f'(a⁺) = lim_{h→0} [f(a+h) – f(a)] / h (h > 0, h → 0)

f(x) is differentiable at x = a iff LHD = RHD (finite and equal). Their common value is f'(a).

Relation between Continuity and Differentiability (from PDF — 4 key results for JEE Main):

1. Differentiability ⟹ Continuity: if f(x) is differentiable at x = a, then f(x) is continuous at x = a.

2. Converse is FALSE: Continuity ⟹ Differentiability is NOT true. f(x) = |x| is continuous everywhere but not differentiable at x = 0.

3. f(x) not differentiable at x = a ⟹ f(x) may or may not be continuous at x = a.

4. f(x) not continuous at x = a ⟹ f(x) NOT differentiable at x = a.

5. From PDF: If LHD and RHD at x = a both exist finitely (even if not equal), then f(x) is continuous at x = a.

Geometric interpretation of differentiability (from PDF): If f(x) is continuous but not differentiable at x = a, it implies a sharp corner or kink at x = a (the tangent is not unique — left and right tangents exist but have different slopes). Examples: y = |x| has a kink at x = 0 (LHD = –1, RHD = +1 ≠ LHD → not differentiable). y = |x²–1| has kinks at x = ±1.

Properties of Differentiable Functions (from PDF — Limits, Continuity and Differentiability):

1. Polynomial, exponential, and constant functions are differentiable at every point of ℝ.

2. Logarithmic and trigonometric functions are differentiable in their domains.

3. Sum, difference, product of differentiable functions is differentiable.

4. Composition of differentiable functions is differentiable (chain rule applies).

5. If f(x) and g(x) are both NOT differentiable at a, their sum f(x)+g(x) and product f(x)·g(x) may still be differentiable at a. (e.g., f(x) = |x| and g(x) = –|x| — both not differentiable at 0, but f+g = 0 IS differentiable.) Download the Free PDF for Limits, Continuity and Differentiability for all differentiability analysis examples for JEE Main.

All 20 Basic Differentiation Formulas — Direct from the PDF

Why These 20 Differentiation Formulas Are the Most-Used Results in JEE Main Maths

The following 20 basic differentiation formulas from the Aakash PDF are the building blocks of all differentiation in JEE Main Maths. Every application of the product rule, chain rule, or implicit differentiation uses these as its base.

20 Basic Differentiation Formulas (from PDF — Limits, Continuity and Differentiability):

1. d/dx (sin x) = cos x

2. d/dx (cos x) = –sin x

3. d/dx (tan x) = sec²x

4. d/dx (cot x) = –cosec²x

5. d/dx (sec x) = sec x tan x

6. d/dx (cosec x) = –cosec x cot x

7. d/dx (eˣ) = eˣ

8. d/dx (log_e x) = 1/x (x > 0)

9. d/dx (aˣ) = aˣ · ln a (a > 0)

10. d/dx (sin⁻¹x) = 1/√(1–x²), for –1 < x < 1

11. d/dx (cos⁻¹x) = –1/√(1–x²), for –1 < x < 1

12. d/dx (tan⁻¹x) = 1/(1+x²), for x ∈ ℝ

13. d/dx (cot⁻¹x) = –1/(1+x²), for x ∈ ℝ

14. d/dx (cosec⁻¹x) = –1/(|x|√(x²–1)), for |x| > 1

15. d/dx (sec⁻¹x) = 1/(|x|√(x²–1)), for |x| > 1

16. d/dx (constant) = 0

17. d/dx (xⁿ) = nxⁿ⁻¹, for n ∈ ℝ

18. d/dx (x⁻ⁿ) = –n/x^(n+1) for x ≠ 0 (from PDF, consequence of formula 17)

19. d/dx (log_a x) = (1/x) · log_a e = 1/(x · ln a) for x > 0, a > 0, a ≠ 1

20. d/dx |x| = x/|x| = sgn(x), for x ≠ 0; undefined at x = 0. (from PDF)

Basic differentiation rules (from PDF — Limits, Continuity and Differentiability):

Scalar multiple: d/dx[c·f(x)] = c·f'(x)

Sum/Difference: d/dx[f(x) ± g(x)] = f'(x) ± g'(x)

Product rule: d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)

Quotient rule: d/dx[f(x)/g(x)] = [f'(x)·g(x) – f(x)·g'(x)] / [g(x)]², g(x) ≠ 0

Download the Free PDF for Limits, Continuity and Differentiability for all 20 basic differentiation formulas with examples for JEE Main.

Chain Rule, Parametric, Implicit, and Logarithmic Differentiation — From PDF

Why These Four Advanced Differentiation Techniques Are Core JEE Main Questions

Chain Rule (from PDF — Limits, Continuity and Differentiability): If y = f(g(x)), then dy/dx = f'(g(x))·g'(x). In terms of intermediate variable: if y = f(t) and t = g(x), then dy/dx = (dy/dt)·(dt/dx). Extended chain rule from PDF: if y = f(g(h(x))), then dy/dx = f'(g(h(x))) · g'(h(x)) · h'(x). The chain rule is the single most-used differentiation tool in JEE Main — every composite function differentiation uses it.

Parametric Differentiation (from PDF — Limits, Continuity and Differentiability): If x = φ(t) and y = ψ(t) (parameter t, φ(t) ≠ 0), then: dy/dx = (dy/dt)/(dx/dt) = ψ'(t)/φ'(t). From PDF remark: d²y/dx² ≠ (d²y/dt²)/(d²x/dt²). The correct formula is: d²y/dx² = d/dx(dy/dx) = [d/dt(dy/dx)]/(dx/dt) = [(d/dt(ψ'(t)/φ'(t)))]/φ'(t).

Logarithmic Differentiation (from PDF — Limits, Continuity and Differentiability): Used for two types: (1) Products and quotients of many functions. (2) f(x)^g(x) type where both base and exponent are functions of x. Method: take ln of both sides, differentiate, solve for dy/dx. From PDF direct rule for [f(x)]^g(x): dy/dx = [f(x)]^g(x) · [g'(x)·ln f(x) + g(x)·f'(x)/f(x)].

Trigonometric substitutions for differentiation (from PDF — Limits, Continuity and Differentiability JEE Main):

1. √(a²–x²) → put x = a sinθ or a cosθ

2. √(a²+x²) → put x = a tanθ or a cotθ

3. √(x²–a²) → put x = a secθ or a cosecθ

4. (a–x)/(a+x) or √((a–x)/(a+x)) → put x = a cos2θ

5. (x–a)/(x+a) → put x = a sec2θ

6. √((a–x)(x–β)) for β < x < a → put x = a sin²θ (from PDF)

7. 2ax–x² = a²–(x–a)² → put x–a = a sinθ

8. (x–α)(β–x) for α < x < β → put x = α cos²θ + β sin²θ (from PDF)

Implicit Differentiation (from PDF — Limits, Continuity and Differentiability): If f(x, y) = 0 defines y as a function of x, differentiate both sides w.r.t. x treating y as a function of x (using chain rule for y-terms), then solve for dy/dx. From PDF note: dy/dx = –(∂f/∂x)/(∂f/∂y) where ∂f/∂x = differentiate treating y constant, ∂f/∂y = differentiate treating x constant.

Differentiation of a function with respect to another function (from PDF): To find d[f(x)]/d[g(x)]: compute f'(x) = df/dx and g'(x) = dg/dx, then d[f(x)]/d[g(x)] = f'(x)/g'(x). This is similar to parametric differentiation.

Differentiation of a determinant (from PDF — Limits, Continuity and Differentiability): If each row of a determinant contains functions of x, differentiate the determinant by differentiating one row at a time (keeping other rows constant) and summing the resulting determinants. Download the Free PDF for all advanced differentiation technique examples for JEE Main.

Derivative of Infinite Series, Higher Order Derivatives, and Leibnitz Formula

Why Higher Order Derivatives and Leibnitz Formula Are JEE Main Limits, Continuity and Differentiability Formula Questions

Derivative of infinite series (from PDF — Limits, Continuity and Differentiability):

Type 1: y = √(g(x) + √(g(x) + √(g(x) + …))). From PDF: express as y = √(g(x) + y) → y² – y = g(x) → differentiate: (2y–1)·dy/dx = g'(x) → dy/dx = g'(x)/(2y–1).

Type 2: y = [g(x)]^([g(x)]^([g(x)]^…)) = [g(x)]^y. From PDF: take log → y·ln g(x) = y·ln g(x)... differentiate: dy/dx = y²·g'(x) / [g(x)·(1 – y·ln g(x))].

Higher Order Derivatives (from PDF — Limits, Continuity and Differentiability): dy/dx = f'(x) is the 1st derivative. d²y/dx² = f''(x) is the 2nd derivative (= d/dx of f'(x)). d³y/dx³ = f'''(x), d⁴y/dx⁴ = f^(iv)(x), etc. Linear combination rule from PDF: (c₁u + c₂v)^(n) = c₁u^(n) + c₂v^(n) where u^(n) = nth derivative.

Leibnitz Formula for nth order derivative of product (from PDF — key JEE Main formula):

(uv)^(n) = Σ_{r=0}^{n} C(n,r) · u^(n–r) · v^(r)

= u^(n)·v + n·u^(n–1)·v' + n(n–1)/2!·u^(n–2)·v'' + … + n·u'·v^(n–1) + u·v^(n)

where u^(0) = u, v^(0) = v, and C(n,r) = n!/(r!(n–r)!) are the binomial coefficients.

nth order derivative formulas (from PDF — Limits, Continuity and Differentiability):

(xᵐ)^(n) = m(m–1)…(m–n+1)·x^(m–n) = ᵐPₙ·x^(m–n); if m = n then (xⁿ)^(n) = n!

(aˣ)^(n) = aˣ·(ln a)ⁿ (a > 0)

(eˣ)^(n) = eˣ (nth derivative of eˣ is eˣ)

(ln x)^(n) = (–1)^(n–1) · (n–1)! / xⁿ (from PDF)

(sin x)^(n) = sin(x + nπ/2)

(cos x)^(n) = cos(x + nπ/2)

Functional equation determining f(x) (from PDF — Limits, Continuity and Differentiability): From PDF example: if f(x+y) = f(x)·f(y), f'(0) = k, then f'(x) = k·f(x) → dy/dx = ky → f(x) = e^(kx). If f(0) = 1: f(x) = e^(kx). Download the Free PDF for all higher order derivative examples and Leibnitz formula applications for JEE Main.