Dabour Hub EGEgyptian Curriculum 2026/2027
1-2 (a) Higher-order derivatives

1-2 (a) Higher-order derivatives

Point 5 of 8 — Implicit Differentiation, Parametric Differentiation and Higher DerivativesQuestion bank: 0 questions (0 with pictures)

05 1-2 (a) Higher-order derivatives

Definition and notation

  • If y=f(x) is differentiable, then y′=dydx=f′(x) is a new function.
  • If y′ is differentiable, its derivative is the second derivative: y′′=d2ydx2=f′′(x).
  • Repeating gives the third derivative y′′′=d3ydx3 and, in general, the nth derivative y(n)=dnydxn=f(n)(x), with n a positive integer.
  • Derivatives from the second onwards are called higher derivatives.

A common confusion

  • d2ydx2 means the derivative of dydx.
  • (dydx)2 means the square of the first derivative.
  • They are different quantities; never mix them.
✍️ Worked examples

Example 11: Find the third derivative of y=x4−2x2+5.

Solution:

  • y′=4x3−4x.
  • y′′=12x2−4.
  • y′′′=24x.

Patterns for the nth derivative

  • y=xn gives y(n)=n!, and every later derivative is 0.
  • y=ekx gives y(n)=knekx.
  • y=sin⁡ax gives y(n)=ansin⁡(ax+nπ2); for y=cos⁡ax use cos⁡(ax+nπ2).
  • Differentiating sin⁡2x four times returns 16sin⁡2x, so the pattern repeats every four steps.

Proving relations between derivatives

  • Differentiate the given relation as many times as the required relation demands, then eliminate constants.
✍️ Worked examples

Example 14: If y=tan⁡x prove that y′′=2y(1+y2).

Solution:

  • y′=sec⁡2x=1+tan⁡2x=1+y2.
  • y′′=2y y′=2y(1+y2).

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