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🔹 1. Exponential & Logarithmic Functions

Function Maclaurin Series Valid for Trick to Remember
\(e^x\) \(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\) All real \(x\) All +, like \(e\) is always positive
\(\ln(1 + x)\) \(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots\) ( x < 1) Alternating signs: \(+\), \(–\), \(+\), \(–\), etc.
\((1 + x)^n\) \(1 + nx + \frac{n(n-1)}{2!}x^2 + \cdots\) ( x < 1) Use binomial expansion

🔹 2. Trigonometric Functions

Function Maclaurin Series Valid for Trick to Remember
\(\sin x\) \(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\) All real \(x\) Odd powers, alternating signs
\(\cos x\) \(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\) All real \(x\) Even powers, alternating signs
\(\tan x\) \(x + \frac{x^3}{3} + \frac{2x^5}{15} + \cdots\) ( x < \frac{\pi}{2}) No easy pattern — memorize first few terms
\(\cot x\) No simple series — Use derivative approach or in terms of \(\tan x\)

🔹 3. Inverse Trigonometric Functions

Function Maclaurin Series Valid for Trick to Remember
\(\arcsin x\) \(x + \frac{x^3}{6} + \frac{3x^5}{40} + \frac{5x^7}{112} + \cdots\) ( x \le 1) All +, only odd powers
\(\arccos x\) \(\frac{\pi}{2} - \arcsin x\) ( x \le 1) Derived from arcsin
\(\arctan x\) \(x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots\) ( x \le 1) Alternating signs, odd powers only

🔹 4. Hyperbolic Functions

Function Maclaurin Series Valid for Trick to Remember
\(\sinh x\) \(x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots\) All real \(x\) Like \(\sin x\), but no alternating signs
\(\cosh x\) \(1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots\) All real \(x\) Like \(\cos x\), but all +

🔹 5. Special Cases / Compositions

Function Notes
\(\sin(x^x)\) Plug \(x^x\) into \(\sin x\) series
\(\sin(x^{1/x})\) Plug \(x^{1/x}\) into \(\sin x\), expand if needed
\(\tan^{-1}(x^2)\) Plug into \(\tan^{-1} x = x - \frac{x^3}{3} + \cdots\)

✅ Memory Tricks (Summary)

Trick Explanation
Sine = Odd, Alt Signs Powers: \(x^1, x^3, x^5...\), signs: \(+\,-\,+\,-\)
Cosine = Even, Alt Signs Powers: \(x^0, x^2, x^4...\), signs: \(+\,-\,+\,-\)
Exponential = All + All powers, all +
ln(1 + x) = Alt, Start at x \(x - x^2/2 + x^3/3 - \cdots\), ( x < 1)
Inverse trig = Odd, Alt or + Like arcsin: all +; arctan: alternate

Complete Differentiation Formula Sheet

I. Basic Derivatives

  1. \(\frac{d}{dx}(c) = 0\) (constant rule)
  2. \(\frac{d}{dx}(x) = 1\)
  3. \(\frac{d}{dx}(x^n) = nx^{n-1}\)
  4. \(\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}\)
  5. \(\frac{d}{dx}(\frac{1}{x}) = -\frac{1}{x^2}\)
  6. \(\frac{d}{dx}(ax + b) = a\)

II. Trigonometric Functions

  1. \(\frac{d}{dx}(\sin x) = \cos x\)
  2. \(\frac{d}{dx}(\cos x) = -\sin x\)
  3. \(\frac{d}{dx}(\tan x) = \sec^2 x\)
  4. \(\frac{d}{dx}(\cot x) = -\csc^2 x\)
  5. \(\frac{d}{dx}(\sec x) = \sec x \tan x\)
  6. \(\frac{d}{dx}(\csc x) = -\csc x \cot x\)

III. Inverse Trigonometric Functions

  1. \(\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}}\)
  2. \(\frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1 - x^2}}\)
  3. \(\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1 + x^2}\)
  4. \(\frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1 + x^2}\)
  5. \(\frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2 - 1}}\)
  6. \(\frac{d}{dx}(\csc^{-1} x) = -\frac{1}{|x|\sqrt{x^2 - 1}}\)

IV. Exponential and Logarithmic Functions

  1. \(\frac{d}{dx}(e^x) = e^x\)
  2. \(\frac{d}{dx}(a^x) = a^x \ln a\)
  3. \(\frac{d}{dx}(\ln x) = \frac{1}{x}\)
  4. \(\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}\)

V. Rules of Differentiation

  1. Sum Rule: \(\frac{d}{dx}(f + g) = f' + g'\)
  2. Difference Rule: \(\frac{d}{dx}(f - g) = f' - g'\)
  3. Product Rule: \(\frac{d}{dx}(fg) = f'g + fg'\)
  4. Quotient Rule: \(\frac{d}{dx}(\frac{f}{g}) = \frac{f'g - fg'}{g^2}\)
  5. Chain Rule: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)

VI. Higher Order Derivatives

  1. \(f''(x) = \frac{d^2f}{dx^2}\)
  2. \(f^{(n)}(x) = \frac{d^n f}{dx^n}\)

VII. Parametric Differentiation

If \(x = f(t), y = g(t)\), then:
\(\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\)

VIII. Implicit Differentiation

If \(y\) is defined implicitly in terms of \(x\):

  • Differentiate both sides w.r.t. \(x\) using chain rule for \(y\)
  • Solve for \(\frac{dy}{dx}\)

IX. Logarithmic Differentiation

For functions like \(y = f(x)^{g(x)}\), take log on both sides:
\(\ln y = g(x) \ln f(x)\)
Then differentiate using product and chain rules.

X. Differentiation of Special Forms

  1. \(y = x^x \Rightarrow \ln y = x \ln x \Rightarrow \frac{dy}{dx} = x^x (\ln x + 1)\)
  2. \(y = a^{\sin x} \Rightarrow \frac{dy}{dx} = a^{\sin x} \ln a \cos x\)

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