Derivative

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Trigonometric functions
Hyperbolic functions
Logarithm
Conics
Vector

Derivative
Integral
Series
Ordinary differential equation

Definition

$$f' = \lim_{\Delta x\rightarrow 0}\frac{f(x + \Delta x) - f(x)}{\Delta x}$$

Identities

$$\begin{array}{ll} \frac{d}{dx}\,a = 0 & \frac{d}{dx}\,(f(x) + g(x)) = f' + g' \\ \frac{d}{dx}\,a \cdot f(x) = a f' & \frac{d}{dx}\,(f(x) - g(x)) = f' - g' \\ \frac{d}{dx}\,f(x)^a = n \, f^{a - 1} f' & \frac{d}{dx}\,f(x) \cdot g(x) = f' g + f g' \\ \frac{d}{dx}\,g(f(x)) = g'(f) \cdot f' & \frac{d}{dx}\,\frac{f(x)}{g(x)} = \frac{f' g - f g'}{g^2} \\[12pt] \frac{d}{dx}\,\sin f(x) = \cos f \cdot f' & \frac{d}{dx}\,\sin^{-1} f(x) = \frac{1}{\sqrt{1 - f^2}} f' \\ \frac{d}{dx}\,\cos f(x) = -\sin f \cdot f' & \frac{d}{dx}\,\cos^{-1} f(x) = \frac{-1}{\sqrt{1 - f^2}} f' \\ \frac{d}{dx}\,\tan f(x) = \frac{1}{\cos^2 f} f' & \frac{d}{dx}\,\tan^{-1} f(x) = \frac{1}{1 + f^2} f' \\ \frac{d}{dx}\,\cot f(x) = \frac{-1}{\sin^2 f} f' & \frac{d}{dx}\,\cot^{-1} f(x) = \frac{-1}{1 + f^2} f' \\[12pt] \frac{d}{dx}\,\ln f(x) = \frac{1}{f} f' & \frac{d}{dx}\,e^{f(x)} = e^f f' \\ \frac{d}{dx}\,\log_a f(x) = \frac{1}{f\ln a} f' & \frac{d}{dx}\,a^{f(x)} = a^f\ln a \cdot f' \\[12pt] \frac{d}{dx}\,\sinh f(x) = \cosh f \cdot f' & \frac{d}{dx}\,\operatorname{sech} f(x) = -\operatorname{sech} f \tanh f \cdot f' \\ \frac{d}{dx}\,\cosh f(x) = \sinh f \cdot f' & \frac{d}{dx}\,\tanh f(x) = \operatorname{sech}^2 f \cdot f' \\ \end{array}$$