Successive differentiation

1. Definition

Successive differentiation is the process of differentiating a function again and again, and the results are the successive derivatives. The practical aim of this topic is to obtain a general formula for the nth derivative yₙ = dⁿy/dxⁿ of a given function, without differentiating n times by hand.

2. The Standard nth Derivatives (must be memorised)

y	yₙ = dⁿy/dxⁿ	Condition
e^(ax)	aⁿ e^(ax)	all n
a^(mx)	aᵐˣ (log a)ⁿ	all n
sin(ax + b)	aⁿ sin(ax + b + nπ/2)	all n
cos(ax + b)	aⁿ cos(ax + b + nπ/2)	all n
(ax + b)^m	[m!/(m − n)!] · aⁿ (ax + b)^(m − n)	m ≥ n
(ax + b)^m	0	m < n (integer m)
(ax + b)^(−m)	(−1)ⁿ · m(m+1)…(m+n−1) · aⁿ · (ax + b)^(−m−n)	negative index
log(ax + b)	(−1)^(n−1) (n − 1)! aⁿ / (ax + b)ⁿ	all n ≥ 1
1/(ax + b)	(−1)ⁿ n! aⁿ / (ax + b)^(n+1)	all n

Why the sin/cos rule works: differentiating sin(ax + b) gives a cos(ax + b) = a sin(ax + b + π/2). Each further differentiation multiplies by a and adds π/2 to the phase, so after n steps the factor is aⁿ and the phase is ax + b + nπ/2.

3. Method for Rational Functions — Partial Fractions

A rational function is first split into partial fractions of the simple forms A/(ax + b) or A/(ax + b)^m, and then the table above is applied to each fraction and the results added.

4. Method for Powers and Products of Trigonometric Functions

Use identities to convert the function into a sum of sines and cosines of multiples of x:

sin²x = (1 − cos 2x)/2, cos²x = (1 + cos 2x)/2, sin³x = (3 sin x − sin 3x)/4, cos³x = (3 cos x + cos 3x)/4;
products: 2 sin A sin B = cos(A − B) − cos(A + B), 2 cos A cos B = cos(A − B) + cos(A + B), 2 sin A cos B = sin(A + B) + sin(A − B).
Then apply the standard sin/cos result term by term. If the function is a product of two different types of function (polynomial × trig, polynomial × exponential), use Leibnitz's theorem instead (next topic).

   Find the nth derivative of   y = x / [(x - 1)(x - 2)]

   Partial fractions:   x/[(x-1)(x-2)] = A/(x-1) + B/(x-2)
        A = [x/(x-2)] at x = 1  = 1/(1-2) = -1
        B = [x/(x-1)] at x = 2  = 2/(2-1) =  2
   ⇒   y = -1/(x - 1) + 2/(x - 2)

   Apply  yₙ of 1/(ax + b)  =  (-1)ⁿ n! aⁿ / (ax + b)^(n+1)  with a = 1 :

        yₙ = -(-1)ⁿ n!/(x - 1)^(n+1) + 2(-1)ⁿ n!/(x - 2)^(n+1)
           = (-1)ⁿ n! [ 2/(x - 2)^(n+1) - 1/(x - 1)^(n+1) ]

   Check for n = 1 :  y′ = 2/(x-2)² - 1/(x-1)²
   Direct differentiation of -1/(x-1) + 2/(x-2) gives 1/(x-1)² - 2/(x-2)²
   ... sign check: d/dx[-1/(x-1)] = +1/(x-1)² and d/dx[2/(x-2)] = -2/(x-2)².
   The formula above with (-1)¹ = -1 gives -[2/(x-2)² - 1/(x-1)²] = 1/(x-1)² - 2/(x-2)²  ✓

8. Choosing the Right Method — Decision Table

Form of y	Method
e^(ax), a^(mx), sin(ax+b), cos(ax+b), (ax+b)^m, log(ax+b), 1/(ax+b)	Read straight from the standard table
Rational function P(x)/Q(x)	Partial fractions, then the 1/(ax+b) or power rule
sin²x, cos²x, sin³x, cos³x, sin ax sin bx, cos ax cos bx, sin ax cos bx	Convert to a sum of sines/cosines of multiples of x
Polynomial × (exponential / trig / log) — e.g. x²eˣ, x³sin 2x, x²log x	Leibnitz's theorem (next topic)

9. Exam Points

Find the nth derivative of a given function using the standard results — 5-7 marks, the most frequent question of this topic.
Find yₙ for a rational function (partial fractions first) — 5-7 marks, very frequent.
Find yₙ for sin²x, cos²x or a product of sines/cosines — 5 marks, frequent.
State the nth derivative of sin(ax + b) and cos(ax + b) — 3 marks.
Show that yₙ = 0 for n > m when y is a polynomial of degree m — 3 marks.
Use successive differentiation to verify a given formula for yₙ — 5 marks.

Successive differentiation