Differentiation — Complete Formula

Basic Definition of Derivative

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Derivative at [x = a]	[f'(a) = \lim_{h \to 0} \dfrac{f(a+h) – f(a)}{h}]	[f'(a)] = derivative at point, [h] = small increment	Fundamental definition ⭐
Alternate Form	[f'(a) = \lim_{x \to a} \dfrac{f(x) – f(a)}{x – a}]	—	Used in limits ⭐

💡 Memory Hint:
Derivative = instantaneous rate of change

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Condition	[f'(-a) = f'(a)]	Left derivative & right derivative	Must be equal ⭐
Relation with Continuity	Differentiable ⇒ Continuous	—	Converse not always true ⚠️

💡 Memory Hint:
Differentiability ⊂ Continuity

Concept	Formula	Symbols Meaning	Key Notes
LHD	[f'(-a) = \lim_{h \to 0^-} \dfrac{f(a-h) – f(a)}{-h}]	Left derivative	Approach from left
RHD	[f'(a) = \lim_{h \to 0^+} \dfrac{f(a+h) – f(a)}{h}]	Right derivative	Approach from right

Function	Formula	Symbols Meaning	Key Notes
Constant	[\dfrac{d}{dx}(c) = 0]	[c] = constant	No change ⭐
Power	[\dfrac{d}{dx}(x^n) = nx^{n-1}]	[n] = real number	Most used ⭐
Exponential	[\dfrac{d}{dx}(e^x) = e^x]	—	Unique property ⭐
Logarithmic	[\dfrac{d}{dx}(\ln x) = \dfrac{1}{x}]	—	Domain [x > 0]
Sine	[\dfrac{d}{dx}(\sin x) = \cos x]	—	Important ⭐
Cosine	[\dfrac{d}{dx}(\cos x) = -\sin x]	—	Sign matters ⚠️
Tangent	[\dfrac{d}{dx}(\tan x) = \sec^2 x]	—	Frequently used

Concept	Formula	Symbols Meaning	Key Notes
Sum	[(f + g)’ = f’ + g’]	[f, g] = functions	Linear property ⭐
Difference	[(f – g)’ = f’ – g’]	—	Same idea

Concept	Formula	Symbols Meaning	Key Notes
Product	[(fg)’ = f’g + fg’]	—	Very important ⭐

💡 Memory Hint:
First derivative × second + first × second derivative

Concept	Formula	Symbols Meaning	Key Notes
Quotient	[\left(\dfrac{f}{g}\right)’ = \dfrac{g f’ – f g’}{g^2}]	—	Numerator order matters ⚠️

💡 Memory Hint:
“Low D high − high D low”

Concept	Formula	Symbols Meaning	Key Notes
Chain Rule	[\dfrac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)]	—	Composite function ⭐

💡 Memory Hint:
Outer derivative × inner derivative

Concept	Formula	Symbols Meaning	Key Notes
Method	Differentiate treating [y] as function of [x]	—	Solve for [\dfrac{dy}{dx}] ⭐

💡 Memory Hint:
Differentiate both sides → isolate [dy/dx]

Concept	Formula	Symbols Meaning	Key Notes
Method	Take log of both sides before differentiation	—	Useful for powers ⭐

💡 Memory Hint:
Complex power → take log first

Function	Formula	Symbols Meaning	Key Notes
[\sin^{-1} x]	[\dfrac{1}{\sqrt{1 – x^2}}]	—	Domain matters ⭐
[\cos^{-1} x]	[-\dfrac{1}{\sqrt{1 – x^2}}]	—	Negative sign ⚠️
[\tan^{-1} x]	[\dfrac{1}{1 + x^2}]	—	Very common ⭐

Function	Condition	Symbols Meaning	Key Notes
Modulus	Not differentiable at [x = 0]	—	Sharp corner ⭐
Greatest Integer	Not differentiable at integers	—	Jump discontinuity
Fractional Part	Not differentiable at integers	—	Same reason

💡 Memory Hint:
Corner / jump → not differentiable

Concept	Formula	Symbols Meaning	Key Notes
Slope of Tangent	[m = \dfrac{dy}{dx}]	—	Instantaneous slope ⭐
Equation of Tangent	[y – y_1 = m(x – x_1)]	[(x_1, y_1)] = point	Direct application ⭐

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