Higher Order Derivatives — Complete Formula

Basic Definition

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Second Derivative	[f”(x) = \dfrac{d}{dx}(f'(x))]	[f”(x)] = second derivative	Rate of change of slope ⭐
nth Derivative	[f^{(n)}(x) = \dfrac{d^n y}{dx^n}]	[n] = order of derivative	General form ⭐

💡 Memory Hint:
Differentiate repeatedly

Concept	Formula	Symbols Meaning	Key Notes
nth Derivative	[\dfrac{d^n}{dx^n}(x^m) = m(m-1)(m-2)\cdots(m-n+1)x^{m-n}]	[m] = power	Stops if [n > m] ⭐

Concept	Formula	Symbols Meaning	Key Notes
[e^x]	[\dfrac{d^n}{dx^n}(e^x) = e^x]	—	Same function ⭐
[a^x]	[\dfrac{d^n}{dx^n}(a^x) = a^x (\ln a)^n]	[a] = constant	Important ⭐

Function	Formula	Symbols Meaning	Key Notes
[\sin x]	Cyclic every 4 derivatives	—	Pattern repeats ⭐
[\cos x]	Cyclic every 4 derivatives	—	Important ⭐

💡 Memory Hint:
sin → cos → −sin → −cos → repeat

Concept	Formula	Symbols Meaning	Key Notes
Leibniz Rule	[(uv)^{(n)} = \sum_{k=0}^{n} \binom{n}{k} u^{(n-k)} v^{(k)}]	[u, v] = functions, [\binom{n}{k}] = binomial coefficient	Very important ⭐

💡 Memory Hint:
Combination-based expansion (like binomial theorem)

Concept	Formula	Symbols Meaning	Key Notes
Form	[e^{ax}(\sin bx \text{ or } \cos bx)]	[a, b] = constants	Use cyclic pattern ⭐

Concept	Formula	Symbols Meaning	Key Notes
[\ln x]	[\dfrac{d^n}{dx^n}(\ln x) = (-1)^{n-1}(n-1)! x^{-n}]	—	Very important ⭐

Concept	Formula	Symbols Meaning	Key Notes
[\tan^{-1} x]	Pattern-based derivatives	—	Requires practice ⭐

Concept	Formula	Symbols Meaning	Key Notes
Second Derivative	[\dfrac{d^2y}{dx^2} = \dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)]	—	Use chain rule ⭐

💡 Memory Hint:
Differentiate again using implicit form

Concept	Formula	Symbols Meaning	Key Notes
First Derivative	[\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}]	[t] = parameter	Basic ⭐
Second Derivative	[\dfrac{d^2y}{dx^2} = \dfrac{d}{dt}\left(\dfrac{dy}{dx}\right) \div \dfrac{dx}{dt}]	—	Important ⭐

Concept	Formula	Symbols Meaning	Key Notes
Concavity	[f”(x) > 0 \Rightarrow \text{concave up}]	—	Curve shape ⭐
Point of Inflection	[f”(x) = 0]	—	Check sign change ⭐

💡 Memory Hint:
Second derivative → curvature

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