Differentiation Formulas

Calculus Tool Reference

1. Fundamental Rules

Rule Name Formula
1. Constant Rule $$ \frac{d}{dx}(C) = 0 $$
2. Constant Multiple $$ \frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x) $$
3. Sum/Difference $$ \frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x) $$
4. Product Rule $$ \frac{d}{dx}(uv) = u’v + uv’ $$
5. Quotient Rule $$ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u’v – uv’}{v^2} $$

2. Algebraic & Exponential

Power Rule ($$ x^n $$)
$$ nx^{n-1} $$
Exponential ($$ e^x $$)
$$ e^x $$
General Exp ($$ a^x $$)
$$ a^x \ln a $$
Log ($$ \ln x $$)
$$ \frac{1}{x} $$

3. Trigonometric Functions

Function Derivative
$$ \sin x $$ $$ \cos x $$
$$ \cos x $$ $$ -\sin x $$
$$ \tan x $$ $$ \sec^2 x $$
$$ \cot x $$ $$ -\csc^2 x $$
$$ \sec x $$ $$ \sec x \tan x $$
$$ \csc x $$ $$ -\csc x \cot x $$

*Hint: All functions starting with ‘c’ have a negative derivative.

4. Inverse Trigonometric

$$ \sin^{-1} x $$: $$ \frac{1}{\sqrt{1-x^2}} $$
$$ \cos^{-1} x $$: $$ \frac{-1}{\sqrt{1-x^2}} $$
$$ \tan^{-1} x $$: $$ \frac{1}{1+x^2} $$
$$ \cot^{-1} x $$: $$ \frac{-1}{1+x^2} $$
$$ \sec^{-1} x $$: $$ \frac{1}{|x|\sqrt{x^2-1}} $$
$$ \csc^{-1} x $$: $$ \frac{-1}{|x|\sqrt{x^2-1}} $$

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