Derivative Rules

Here we provide the basic derivative rules. We separated them into (1) the most basic identities you should memorize, and (2) properties of the derivatives of complex functions.

1) Identities

• \(f(x) = c \implies f'(x) = 0\),
• \(f(x) = x^n \implies f'(x) = nx^{n-1}\),
• \(f(x) = a^x \implies f'(x) = a^x \log a\), and hence \(f(x) = e^x \implies f'(x) = e^x\),
• \(f(x) = \log_b x \implies f'(x) = \frac{1}{\log (b) \cdot x}\), and hence \(f(x) = \log x \implies f'(x) = \frac{1}{x}\),
• \(f(x) = \sin x \implies f'(x) = \cos x\),
• \(f(x) = \cos x \implies f'(x) = - \sin x\),
• \(f(x) = \tan x \implies f'(x) = \frac{1}{\cos^2 x}\).

Moreover, it is useful to remember special cases of the second rule:

• \(f(x) = x \implies f'(x) = 1\),
• \(f(x) = ax \implies f'(x) = a\),
• \(f(x) = \sqrt{x} \implies f'(x) = \frac{1}{2\sqrt{x}}\).

2) Complex derivative rules

• \((c \cdot f)'(x) = c \cdot f'(x)\),
• \((f + g)'(x) = f'(x) + g'(x)\),
• \((f \cdot g)'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)\),
• \((\frac{f}{g})'(x) = \frac{f'(x) g(x) - f(x)g'(x)}{g(x)^2}\),
• \((f \circ g)'(x) = (f' \circ g)(x) \cdot g'(x)\).