Higher order derivatives

Higher order derivatives

1. Notation

Differentiating a function repeatedly gives its higher order derivatives:

Order	Leibniz notation	Prime notation	Operator
1st	dy/dx	y′, f′(x)	Dy
2nd	d²y/dx²	y″, f″(x)	D²y
3rd	d³y/dx³	y‴, f‴(x)	D³y
nth	dⁿy/dxⁿ	y⁽ⁿ⁾, f⁽ⁿ⁾(x)	Dⁿy

The second derivative is obtained by differentiating the first derivative, the third by differentiating the second, and so on.

2. What the Higher Derivatives Mean

• First derivative f′ — instantaneous rate of change (slope of the tangent).
• Second derivative f″ — rate of change of the slope. This is the concavity: f″ > 0 → the curve bends upward (convex), f″ < 0 → bends downward (concave). It drives the second-derivative test for maxima and minima (Module 3) and the sign of acceleration in mechanics.
• Mechanics: if s(t) is displacement, then v = ds/dt is velocity, a = d²s/dt² is acceleration, and d³s/dt³ is jerk.
• Third and higher orders appear in Taylor and Maclaurin series (Module 3), in curvature, and in the solution of differential equations.

3. How to Compute Them — Practical Rules

1. Differentiate term by term; use the product, quotient and chain rules at every step.
2. Simplify before differentiating again — that is the whole trick. For example, a product such as x eˣ gives a clean pattern if you factor eˣ out each time.
3. For rational functions, split into partial fractions first (next topic), and for powers of trig functions, reduce with identities such as sin²x = (1 − cos 2x)/2.
4. Verify with the pattern: polynomial of degree m → the (m + 1)th and higher derivatives are zero.

   y = x⁴ + 3x³ - 2x² + 5x - 7

   y′  = 4x³ + 9x² - 4x + 5
   y″  = 12x² + 18x - 4
   y‴  = 24x + 18
   y⁗  = 24
   y⁵  = 0      and therefore every higher derivative is 0
   (degree 4 ⇒ five non-zero derivatives at most)

7. Standard nth-Derivative Results (found once, used everywhere)

y	dⁿy/dxⁿ
e^(ax)	aⁿ e^(ax)
a^(mx)	aᵐˣ (log a)ⁿ
sin(ax + b)	aⁿ sin(ax + b + nπ/2)
cos(ax + b)	aⁿ cos(ax + b + nπ/2)
(ax + b)^m (m ≥ n)	[m!/(m − n)!] aⁿ (ax + b)^(m − n)
log(ax + b)	(−1)^(n − 1) (n − 1)! aⁿ / (ax + b)ⁿ
1/(ax + b)	(−1)ⁿ n! aⁿ / (ax + b)^(n + 1)
e^(ax) sin(bx + c)	(a² + b²)^(n/2) e^(ax) sin(bx + c + nθ), θ = arctan(b/a)

The sin/cos entries come from the phase-shift trick: differentiating sin(ax + b) once gives a cos(ax + b) = a sin(ax + b + π/2), and each further differentiation adds another π/2 to the phase. These results are derived and applied in the next topic.

8. Exam Points

• Define the second and nth order derivatives; write the notation dy/dx, d²y/dx², dⁿy/dxⁿ — 3 marks.
• Find the second and third derivatives of a given function (polynomial, trig, exponential, product) — 5 marks, almost certain.
• Interpret the second derivative physically (acceleration) and geometrically (concavity) — 3-5 marks, frequent.
• Given s(t), find velocity, acceleration and the instants when velocity is zero — 5 marks, frequent.
• Show that all derivatives of order > m of a degree-m polynomial vanish — 3 marks.
• State the four-cycle property of sin x / cos x derivatives — 3 marks.