(1) ln equations
$$\frac{d}{dx} \ln(x) = \frac{1}{x}$$
$$\frac{d}{dx} \ln(f(x)) = \frac{f'(x)}{f(x)}$$
Example:
$$\frac{d}{dx} \ln(3x^3 + 6x) = \frac{9x^2 + 6}{3x^3 + 6x} = \frac{3x^2 + 2}{x^2 + 2x}$$
(2) e equations
$$\frac{d}{dx} e^x = e^x$$
$$\frac{d}{dx} e^{f(x)} = f'(x) e^{f(x)}$$
Example: \( e^{3x^2 + 2} = (6x) e^{3x^2 + 2} \)
(3) Trigonometry
$$\begin{array}{|c|c|} \hline \sin x, \sin(f(x)) & \cos x, f'(x) \cos(f(x)) \\ \hline \cos x, \cos(f(x)) & -\sin x, -f'(x) \sin(f(x)) \\ \hline \tan x, \tan(f(x)) & \tan x, \sec^2 x, f'(x) \sec^2(f(x)) \\ \hline \end{array}$$
Note: If you have to differentiate something like \( \sin^2 2x \) you can use the chain rule and treat it as \( (\sin 2x)^2 \)
O-level/SPM revision for basic differentiation
$$\frac{d}{dx} ax^b = abx^{b-1}$$
Example: \( 3x^4 = 3(4)x^{4-1} = 12x^3 \)
$$\frac{d}{dx} [f(x)]^n = n[f(x)]^{n-1} f'(x)$$
Example: \( (3x^3 + 12)^6 = 6(3x^3 + 12)^5 (9x^2) = 54x^2 (3x^3 + 12)^5 \)