Product, Quotient & Chain Rule

You know what a derivative is. Slope of the tangent. Instantaneous rate of change. The lesson here is about mechanics — when the expression is messy and built from simpler pieces, how do you find the slope without going back to the limit definition every single time?

Three rules cover almost everything GATE will throw at you: product, quotient, and chain. Learn the patterns once and the rest is bookkeeping.

The building blocks

Before the rules, you must know the derivatives of the standard functions cold. These are the atoms; the rules are how you combine them.

In words: d/dx(xⁿ) = n·xⁿ⁻¹, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x, d/dx(sin x) = cos x, and d/dx(cos x) = −sin x. Note the minus sign on cosine — it is the single most-forgotten detail here.

Switch between these functions in the widget below and drag the point along the curve. The number under “slope” is f'(x) at that location — for sin x it reads cos x, for x² it reads 2x. Watching the slope value follow the rule you just memorised is a quick way to make these atoms feel real.

The product rule

When two functions are multiplied, you cannot just multiply their derivatives. Instead, differentiate one factor at a time and add:

(f · g)' = f'·g + f·g'

Differentiate the first, keep the second; then keep the first, differentiate the second; sum the two. (Memorisable as “first-deriv times second, plus first times second-deriv”.)

The quotient rule

For a ratio f/g, the pattern is similar but with a subtraction and a squared denominator:

( f / g )' =  ( f'·g − f·g' ) / g²

The numerator is f'g − fg' — low-d-high minus high-d-low, all over the bottom squared. The order matters: the term with f' (derivative of the top) comes first, and the second term is subtracted.

The chain rule

The chain rule handles a function inside another function — a composition like sin(3x²). Differentiate from the outside in, and multiply by the derivative of the inside:

( f(g(x)) )' = f'(g(x)) · g'(x)

Think of nested layers: peel the outer function (differentiate it, leaving the inside untouched), then multiply by the derivative of the next layer in. Each layer contributes one factor.

How GATE asks this

A NAT or MCQ hands you a function built from the standard pieces and asks for its derivative — either the symbolic form (MCQ: pick the correct expression) or the derivative evaluated at a point (NAT: a single number). The function is deliberately a combination, so you must spot which rule (or rules) applies: a product like x²eˣ, a composition like sin(3x²), or a ratio like x/(x+1). The skill being tested is selecting and applying the right rule cleanly — there is no trick beyond the bookkeeping.

Worked example — applying each rule

(1) Product rule — differentiate h(x) = x² · eˣ.

Here f = x² (so f' = 2x) and g = eˣ (so g' = eˣ):

h'(x) = f'·g + f·g'
      = (2x)·eˣ + (x²)·eˣ
      = (2x + x²) eˣ

(2) Chain rule — differentiate sin(3x²).

Outer is sin(u) with u = 3x². The outer derivative is cos(u) = cos(3x²); the inner derivative is d/dx(3x²) = 6x. Multiply:

d/dx sin(3x²) = cos(3x²) · 6x = 6x · cos(3x²)

The 6x is the inner-derivative factor — drop it and the answer is wrong.

(3) Quotient rule — differentiate (x) / (x + 1).

Top f = x (so f' = 1), bottom g = x + 1 (so g' = 1):

d/dx  x/(x+1) = ( f'·g − f·g' ) / g²
              = ( 1·(x+1) − x·1 ) / (x+1)²
              = ( x + 1 − x ) / (x+1)²
              = 1 / (x+1)²

The numerator collapses to 1, leaving 1/(x+1)² — always positive, which makes sense since x/(x+1) is increasing everywhere it is defined.

Quick check