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Higher Order Derivatives



Definition and Notation
The Second Derivative
Physical Interpretation
Patterns in Repeated Differentiation — Polynomials
Patterns in Repeated Differentiation — Exponentials
Patterns in Repeated Differentiation — Sine and Cosine
The nth Derivative of Specific Forms
Higher-Order Derivatives and Taylor Series
Existence and Smoothness Classes
nth Derivatives at a Glance



Differentiating the Derivative


The derivative f′f' is a function in its own right, and functions can be differentiated. Applying the differentiation process to f′f' produces the second derivative f′′f'', which measures how the slope of ff changes. Differentiating again yields f′′′f''', then f(4)f^{(4)}, and so on without limit—provided the relevant limits exist at each stage.

Each successive derivative captures a finer layer of a function's behavior. The first derivative controls direction. The second controls concavity. The third and beyond govern subtler aspects of curvature and, in physics, correspond to jerk, snap, and higher kinematic quantities. At the theoretical level, the entire collection of higher-order derivatives at a point encodes the local shape of a function through its Taylor series.

Key Terms

Higher-Order Derivative— f(n)(x)f^{(n)}(x), differentiating ff a total of nn times
Derivative— the first-order case from which all higher orders follow
Concavity— governed by the second derivative f′′f''
Inflection Point— where f′′f'' changes sign

See All Calculus Definitions →


The Second Derivative


The second derivative f′′(x)f''(x) measures the rate of change of the slope. Where f′′(x)>0f''(x) > 0, the slope f′(x)f'(x) is increasing—the function bends upward (concave up). Where f′′(x)<0f''(x) < 0, the slope is decreasing—the function bends downward (concave down).

This information is independent of whether ff itself is increasing or decreasing. A function can rise while decelerating (f′>0f' > 0, f′′<0f'' < 0) or fall while accelerating in the negative direction (f′<0f' < 0, f′′<0f'' < 0). The first and second derivatives describe different aspects of behavior.

The second derivative also powers the second derivative test for classifying critical points. At a point where f′(c)=0f'(c) = 0: if f′′(c)>0f''(c) > 0, the critical point is a local minimum; if f′′(c)<0f''(c) < 0, a local maximum; if f′′(c)=0f''(c) = 0, the test is inconclusive.

Inflection points—where concavity reverses—occur where f′′f'' changes sign. The condition f′′(c)=0f''(c) = 0 is necessary but not sufficient; the sign of f′′f'' must actually switch across cc.

Physical Interpretation


In kinematics, the first three derivatives of position s(t)s(t) have standard names.

The first derivative s′(t)s'(t) is velocity: the rate of change of position. It tells how fast an object moves and in which direction.

The second derivative s′′(t)s''(t) is acceleration: the rate of change of velocity. Positive acceleration means speeding up (in the positive direction) or decelerating (in the negative direction). The sign of acceleration relative to the sign of velocity determines whether the object is speeding up or slowing down.

The third derivative s′′′(t)s'''(t) is jerk: the rate of change of acceleration. Jerk is felt physically as a sudden push or lurch—smooth motion has low jerk, while abrupt starts and stops produce high jerk. Elevator design, roller coaster engineering, and vehicle ride comfort all involve controlling jerk.

Beyond the third derivative, the terms snap (s(4)s^{(4)}), crackle (s(5)s^{(5)}), and pop (s(6)s^{(6)}) are used in specialized engineering contexts but rarely appear in standard calculus. The table below collects all six in order, with each name paired with what the derivative measures.
Order Derivative of position Standard name What it measures
1 s'(t) velocity rate of change of position; speed and direction of motion
2 s''(t) acceleration rate of change of velocity; force-related quantity in Newton's second law
3 s'''(t) jerk rate of change of acceleration; felt physically as a sudden push or lurch
4 s(4)(t) snap (or jounce) specialised engineering use; rarely appears in standard calculus
5 s(5)(t) crackle named by analogy with snap; specialised engineering use
6 s(6)(t) pop named by analogy with snap and crackle; specialised engineering use

Patterns in Repeated Differentiation — Polynomials


Polynomials terminate under repeated differentiation. Each differentiation reduces the degree by one, so a polynomial of degree nn reaches a constant after nn differentiations and becomes zero after n+1n + 1.

For f(x)=x5f(x) = x^5:

f′(x)=5x4,f′′(x)=20x3,f′′′(x)=60x2,f(4)(x)=120x,f(5)(x)=120,f(6)(x)=0f'(x) = 5x^4, \quad f''(x) = 20x^3, \quad f'''(x) = 60x^2, \quad f^{(4)}(x) = 120x, \quad f^{(5)}(x) = 120, \quad f^{(6)}(x) = 0


The coefficient at the nnth derivative of xnx^n is n!=n(n−1)(n−2)⋯1n! = n(n-1)(n-2) \cdots 1, accumulated from the power rule applied nn times. Specifically, dndxn[xn]=n!\frac{d^n}{dx^n}[x^n] = n! and dkdxk[xn]=0\frac{d^k}{dx^k}[x^n] = 0 for all k>nk > n.

For a general polynomial p(x)=anxn+⋯+a1x+a0p(x) = a_n x^n + \cdots + a_1 x + a_0, the nnth derivative is n!⋅ann! \cdot a_n, a constant. This relationship becomes central in Taylor series, where the coefficient ana_n is recovered as f(n)(a)n!\frac{f^{(n)}(a)}{n!}.

Patterns in Repeated Differentiation — Exponentials


The natural exponential exe^x is unchanged by differentiation:

nth Derivative of Natural Exponential
dndxn[ex]=ex\frac{d^n}{dx^n}[e^x] = e^x
Learn more about this formula: nth Derivative of Natural Exponential →

Every derivative of exe^x is exe^x. No other elementary function has this property (aside from the trivial f(x)=0f(x) = 0).

For the general exponential eaxe^{ax}, the chain rule introduces a factor of aa at each step:

nth Derivative of Scaled Exponential
dndxn[eax]=aneax\frac{d^n}{dx^n}[e^{ax}] = a^n e^{ax}
Learn more about this formula: nth Derivative of Scaled Exponential →


Each differentiation multiplies by aa. After nn differentiations, the accumulated constant is ana^n. This pattern appears in solutions to differential equations, where eaxe^{ax} satisfies equations whose characteristic root is aa.

For axa^x with arbitrary base: since ax=exln⁡aa^x = e^{x \ln a}, the nnth derivative is (ln⁡a)n⋅ax(\ln a)^n \cdot a^x. The factor ln⁡a\ln a replaces aa in the exponential pattern.

Patterns in Repeated Differentiation — Sine and Cosine


The derivatives of sin⁡x\sin x cycle with period four:

sin⁡x→cos⁡x→−sin⁡x→−cos⁡x→sin⁡x→⋯\sin x \to \cos x \to -\sin x \to -\cos x \to \sin x \to \cdots


The nnth derivative of sin⁡x\sin x depends on nmod  4n \mod 4:

dndxn[sin⁡x]={sin⁡xn≡0(mod4)cos⁡xn≡1(mod4)−sin⁡xn≡2(mod4)−cos⁡xn≡3(mod4)\frac{d^n}{dx^n}[\sin x] = \begin{cases} \sin x & n \equiv 0 \pmod{4} \\ \cos x & n \equiv 1 \pmod{4} \\ -\sin x & n \equiv 2 \pmod{4} \\ -\cos x & n \equiv 3 \pmod{4} \end{cases}


A compact formula captures all four cases: dndxn[sin⁡x]=sin⁡(x+nπ2)\frac{d^n}{dx^n}[\sin x] = \sin\left(x + \frac{n\pi}{2}\right). The same holds for cosine: dndxn[cos⁡x]=cos⁡(x+nπ2)\frac{d^n}{dx^n}[\cos x] = \cos\left(x + \frac{n\pi}{2}\right).

nth Derivative of Sine
dndxn[sin⁡x]=sin⁡ ⁣(x+nπ2)\frac{d^n}{dx^n}[\sin x] = \sin\!\left(x + \frac{n\pi}{2}\right)
Learn more about this formula: nth Derivative of Sine →

nth Derivative of Cosine
dndxn[cos⁡x]=cos⁡ ⁣(x+nπ2)\frac{d^n}{dx^n}[\cos x] = \cos\!\left(x + \frac{n\pi}{2}\right)
Learn more about this formula: nth Derivative of Cosine →


For sin⁡(ax)\sin(ax), the chain rule introduces a factor of aa per differentiation: dndxn[sin⁡(ax)]=ansin⁡(ax+nπ2)\frac{d^n}{dx^n}[\sin(ax)] = a^n \sin\left(ax + \frac{n\pi}{2}\right). The cycle in the trigonometric part persists; only the amplitude grows as ana^n.

This four-fold periodicity distinguishes trigonometric derivatives from polynomial derivatives (which terminate) and exponential derivatives (which replicate).

The nth Derivative of Specific Forms


Several standard functions have known closed-form nnth derivatives.

For f(x)=1xf(x) = \frac{1}{x}: rewriting as x−1x^{-1} and applying the power rule repeatedly gives

nth Derivative of Reciprocal
dndxn ⁣[1x]=(−1)n n!xn+1\frac{d^n}{dx^n}\!\left[\frac{1}{x}\right] = \frac{(-1)^n\, n!}{x^{n+1}}
Learn more about this formula: nth Derivative of Reciprocal →

Each differentiation multiplies by one more negative integer, producing the factorial and the alternating sign.

For f(x)=ln⁡xf(x) = \ln x: since f′(x)=x−1f'(x) = x^{-1}, the higher derivatives follow the pattern above shifted by one:

nth Derivative of Natural Logarithm
dndxn[ln⁡x]=(−1)n−1 (n−1)!xn\frac{d^n}{dx^n}[\ln x] = \frac{(-1)^{n-1}\, (n-1)!}{x^n}
Learn more about this formula: nth Derivative of Natural Logarithm →

For f(x)=xmf(x) = x^m where mm is a positive integer and n≤mn \leq m:

nth Derivative of Power
dndxn[xm]=m!(m−n)! xm−n(n≤m)\frac{d^n}{dx^n}[x^m] = \frac{m!}{(m-n)!}\, x^{m-n} \quad (n \leq m)
Learn more about this formula: nth Derivative of Power →


The coefficient m!(m−n)!\frac{m!}{(m-n)!} is the falling factorial, counting the multipliers accumulated over nn applications of the power rule.

These closed-form expressions are useful for computing specific high-order derivatives without performing each differentiation step individually.

Higher-Order Derivatives and Taylor Series


The Taylor series of ff centered at x=ax = a is

Taylor Series
f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n
Learn more about this formula: Taylor Series →


Each coefficient depends on a higher-order derivative evaluated at the center point aa. The zeroth derivative f(0)(a)=f(a)f^{(0)}(a) = f(a) gives the constant term. The first derivative gives the linear term. The second derivative gives the quadratic correction. Each successive term captures finer detail about how ff deviates from the lower-order approximation.

The Taylor polynomial of degree kk truncates the series after k+1k + 1 terms, providing a polynomial approximation to ff near aa. The quality of the approximation improves with kk—more derivatives mean a closer fit over a wider interval.

The connection between higher-order derivatives and Taylor series gives these derivatives their deepest significance: the complete collection {f(n)(a)}n=0∞\{f^{(n)}(a)\}_{n=0}^{\infty} determines ff locally (for analytic functions). Knowing all derivatives at a single point reconstructs the entire function in a neighborhood of that point.

Existence and Smoothness Classes


A function may be differentiable once but not twice. The function f(x)=x∣x∣f(x) = x|x| has f′(x)=2∣x∣f'(x) = 2|x|, which is continuous but not differentiable at x=0x = 0. So ff is in class C1C^1 (continuously differentiable) but not C2C^2.

The smoothness classes organize functions by how many continuous derivatives they possess. A function belongs to CnC^n if f,f′,f′′,…,f(n)f, f', f'', \ldots, f^{(n)} all exist and are continuous. Class C0C^0 is simply continuous functions. Class C∞C^\infty consists of infinitely differentiable functions—called smooth functions—where derivatives of all orders exist and are continuous.

Polynomials, exe^x, sin⁡x\sin x, cos⁡x\cos x, and their compositions are all C∞C^\infty. Piecewise functions typically belong to a finite CnC^n class determined by how smoothly the pieces join at their boundaries: matching values gives C0C^0, matching first derivatives gives C1C^1, and so on.

There exist functions that are C∞C^\infty but not analytic—their Taylor series converges but not to the function itself. The standard example is f(x)=e−1/x2f(x) = e^{-1/x^2} for x≠0x \neq 0 and f(0)=0f(0) = 0: every derivative at x=0x = 0 is zero, so the Taylor series is identically zero, yet the function is not zero away from the origin. Smooth does not automatically mean representable by a power series. The table below lays out the main tiers — C⁰, the general C&#8319;, C&#8734;, and the strictly stronger analytic class — alongside what each requires and characteristic examples.
Class What it requires Examples
C0 f is continuous (no derivatives required) |x|; piecewise functions whose pieces match values at the boundary
Cn f, f', f'', … , f(n) all exist and are continuous x|x| belongs to C1 but not C2
C∞  (smooth) derivatives of all orders exist and are continuous polynomials, ex, sin x, cos x, and their compositions
Analytic C∞, and the Taylor series at every point converges to f near that point most elementary functions; counterexample f(x) = e−1 ⁄ x² (extended by f(0) = 0) is C∞ but not analytic at 0

nth Derivatives at a Glance


Different function families behave very differently under repeated differentiation: polynomials terminate, exponentials replicate, and trigonometric functions cycle. The table below collects the closed-form nth derivative for each standard function discussed above into a single reference card, with a brief note on what drives the pattern.
Function nth derivative Note
xm  (m a positive integer) m! ⁄ (m − n)! · xm − n  for n ≤ m falling factorial; becomes the constant m! when n = m, and zero for n > m
Polynomial of degree m m! · am  (when n = m);  0 (when n > m) terminates after m + 1 differentiations
1 ⁄ x (−1)n · n! ⁄ xn + 1 alternating sign accompanied by factorial growth in the numerator
ln x (−1)n − 1 · (n − 1)! ⁄ xn  (n ≥ 1) the 1 ⁄ x pattern shifted by one order, since (ln x)' = 1 ⁄ x
ex ex self-replicating; the only nonzero elementary function with this property
ea x an · ea x each differentiation multiplies by a, so the constant accumulates as an
ax (ln a)n · ax since ax = ex · ln a, ln a takes the role of the multiplier
sin x sin(x + n π ⁄ 2) four-fold cycle: sin → cos → −sin → −cos → sin
cos x cos(x + n π ⁄ 2) same four-fold cycle, starting from cosine
sin(a x) an · sin(a x + n π ⁄ 2) trigonometric cycle persists; amplitude grows as an