Introduction

Differentiation is the branch of calculus that measures how a quantity changes. It's one of the most heavily tested topics across JEE, KCET and Board maths papers, and it forms the base for later topics like maxima/minima, tangents, and application-based problems in physics.

What a derivative actually represents

The derivative of a function f(x) at a point tells you the instantaneous rate of change of f with respect to x at that point — geometrically, the slope of the tangent line to the curve at that point. If f(x) is position with respect to time, f′(x) is velocity: how fast position changes at that exact instant.

Definition f'(x) = lim(h→0) [f(x+h) − f(x)] / h

Key differentiation rules

Rules you must know
  • Power rule: d/dx (xⁿ) = n·xⁿ⁻¹
  • Constant rule: d/dx (c) = 0
  • Sum/difference rule: d/dx [f(x) ± g(x)] = f′(x) ± g′(x)
  • Product rule: d/dx [f·g] = f′·g + f·g′
  • Quotient rule: d/dx [f/g] = (f′·g − f·g′) / g²
  • Chain rule: d/dx [f(g(x))] = f′(g(x)) · g′(x)
  • Common derivatives: d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x
Worked example — power rule Differentiate f(x) = 3x⁴ − 5x² + 7.

f′(x) = 3(4x³) − 5(2x) + 0 = 12x³ − 10x
Worked example — product rule Differentiate f(x) = x²·sin x.

f′(x) = (2x)(sin x) + (x²)(cos x) = 2x·sin x + x²·cos x
Worked example — chain rule Differentiate f(x) = sin(3x² + 1).

Let u = 3x² + 1. Then f = sin(u), f′ = cos(u) · u′ = cos(3x² + 1) · 6x = 6x·cos(3x² + 1)

Common mistakes

Watch out for
  • Forgetting the chain rule when differentiating a composite function like sin(3x) — the derivative isn't just cos(3x), you must multiply by the derivative of the inner function.
  • Applying the product rule as if it were "multiply the derivatives" — it isn't; you must use f′·g + f·g′.
  • Dropping constants — d/dx(5x³) = 15x², not 5x².
  • Mixing up d/dx(sin x) = cos x with d/dx(cos x) = −sin x (note the sign).

Exam preparation tips

Tip
  • Memorise the standard derivative table (trig, exponential, log functions) — most problems build on these.
  • Practice identifying which rule applies before you start differentiating; misidentifying product vs. chain rule is the most common exam error.
  • For JEE-level problems, get comfortable combining multiple rules in a single question (e.g., chain rule inside a product rule).
  • Always simplify your final derivative — many marks are lost to unsimplified but technically correct answers.

Summary

A derivative measures instantaneous rate of change — the slope of a tangent at a point. Master the power, product, quotient and chain rules along with standard derivatives, and most differentiation problems in JEE, KCET and Boards become a matter of correctly identifying which rule (or combination of rules) to apply.

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