GATE/Engineering Mathematics/Calculus — Limits, Derivatives & Integration
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Medium18 min readEngineering Mathematics

Calculus — Limits, Derivatives & Integration

Calculus covers limits, differentiation, integration, and differential equations. GATE tests L'Hôpital's rule, Taylor series, partial derivatives, and basic ODEs.

Key Points

  • ·L'Hôpital's rule: lim f/g = lim f'/g' when form is 0/0 or ∞/∞
  • ·Chain rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)
  • ·Taylor/Maclaurin: eˣ = 1+x+x²/2!+..., sin x = x−x³/3!+...
  • ·Integration by parts: ∫u dv = uv − ∫v du (LIATE rule for choosing u)
  • ·Partial derivative: ∂f/∂x treats all other variables as constants
  • ·Separable ODE: dy/dx = f(x)g(y) → separate and integrate both sides

What is a Limit?

Imagine walking towards a wall. As you get closer and closer, the "limit" is where the wall is — even if you never actually touch it.

lim_{x→2} (x² − 4)/(x − 2)

At x=2, this is 0/0 — undefined. But as x gets close to 2:

(x² − 4)/(x − 2) = (x+2)(x−2)/(x−2) = x + 2

As x → 2, this → 4. The limit is 4, even though f(2) is undefined.

L'Hôpital's Rule — The 0/0 Escape Hatch

When a limit gives 0/0 or ∞/∞, take derivatives of top and bottom separately:

lim_{x→0} sin(x)/x    [form: 0/0]
= lim_{x→0} cos(x)/1   [differentiate top and bottom]
= cos(0)/1 = 1

Common limits to memorise (all provable with L'Hôpital):

lim_{x→0} sin(x)/x  = 1
lim_{x→0} (eˣ−1)/x  = 1
lim_{x→0} ln(1+x)/x  = 1
lim_{x→∞} (1+1/x)^x = e

Derivatives — Rate of Change

The derivative tells you the slope of a curve at a point — how fast things are changing.

Rules You Must Know

Rule Formula Example
Power d/dx[xⁿ] = nxⁿ⁻¹ d/dx[x³] = 3x²
Chain d/dx[f(g(x))] = f'(g(x))·g'(x) d/dx[sin(x²)] = cos(x²)·2x
Product d/dx[f·g] = f'g + fg' d/dx[x·eˣ] = eˣ + xeˣ
Quotient d/dx[f/g] = (f'g − fg')/g² d/dx[sin x/x]

Key derivatives:

d/dx[eˣ] = eˣ         d/dx[ln x] = 1/x
d/dx[sin x] = cos x    d/dx[cos x] = −sin x
d/dx[tan x] = sec²x    d/dx[aˣ] = aˣ ln a

Chain Rule — Step by Step

Find d/dx[e^(x²)]:

Let g(x) = x²,  f(u) = eᵘ
f'(u) = eᵘ,  g'(x) = 2x
Answer: e^(x²) · 2x

Taylor and Maclaurin Series — Approximating Functions

Every smooth function can be written as an infinite polynomial around a point.

Maclaurin series (around x = 0):

eˣ   = 1 + x + x²/2! + x³/3! + ...
sin x = x − x³/3! + x⁵/5! − ...     (only odd powers)
cos x = 1 − x²/2! + x⁴/4! − ...     (only even powers)
ln(1+x) = x − x²/2 + x³/3 − ...     (valid for |x| < 1)

Use: evaluate limits without L'Hôpital:

lim_{x→0} (eˣ − 1 − x)/x²
= lim_{x→0} [(1+x+x²/2+...) − 1 − x] / x²
= lim_{x→0} [x²/2 + x³/6 + ...] / x²
= lim_{x→0} [1/2 + x/6 + ...]
= 1/2

Integration — Reversing the Derivative

The definite integral ∫ₐᵇ f(x)dx gives the area under f(x) from a to b.

Fundamental Theorem: ∫ₐᵇ f(x)dx = F(b) − F(a), where F'(x) = f(x).

Integration by Parts

When the integrand is a product, use: ∫u dv = uv − ∫v du

Choose u using LIATE order (L = Logarithm, I = Inverse trig, A = Algebraic, T = Trig, E = Exponential). Pick the first type in LIATE as u.

Example: ∫x·eˣ dx

u = x    (Algebraic comes before Exponential in LIATE)
dv = eˣ dx
du = dx,   v = eˣ

∫x·eˣ dx = x·eˣ − ∫eˣ dx = x·eˣ − eˣ + C = eˣ(x−1) + C

Partial Derivatives

When a function has multiple variables, the partial derivative with respect to x treats all other variables as constants.

f(x, y) = x²y + 3xy² + y³

∂f/∂x = 2xy + 3y²      (treat y as constant, differentiate in x)
∂f/∂y = x² + 6xy + 3y²  (treat x as constant, differentiate in y)

Finding critical points: set both partial derivatives to zero and solve simultaneously.

Differential Equations

A differential equation involves a function and its derivatives.

Separable ODE: dy/dx = f(x)·g(y)

Separate:   dy/g(y) = f(x) dx
Integrate both sides

Example: dy/dx = xy

dy/y = x dx
ln|y| = x²/2 + C
y = Ae^(x²/2)    where A = e^C

Quick Check

Q1. Find lim_{x→0} (sin 3x)/x.

Answer: lim = 3 · lim_{x→0} (sin 3x)/(3x) = 3 × 1 = 3.

Q2. What is d/dx[ln(sin x)]?

Answer: By chain rule: (1/sin x) · cos x = cot x.

Q3. Evaluate ∫₀¹ x·eˣ dx.

Answer: Using IBP result eˣ(x−1) + C: = [eˣ(x−1)]₀¹ = e¹(0) − e⁰(−1) = 0 − (−1) = 1.

Key Formulas

  • L'Hôpital's Rule: lim f/g = lim f'/g' when form is 0/0 or ∞/∞
  • Chain rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)
  • Integration by parts: ∫u dv = uv − ∫v du (LIATE to choose u)
  • Taylor series: f(x) = Σ f^(n)(a)/n! · (x−a)^n

GATE Exam Tips

  • ★eˣ Maclaurin series is used frequently in GATE limit evaluations — memorise 1+x+x²/2!+x³/3!
  • ★L'Hôpital: apply ONLY when limit is 0/0 or ∞/∞ — applying it otherwise is wrong
  • ★Integration by parts: LIATE order for choosing u saves time
  • ★Partial derivatives: treat other variables as numbers — straightforward once you practice

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