APEX Differentiation Formulas

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Section B.1 Differentiation Formulas

List B.1.1. Derivative Rules

$\frac{d}{d x} (c x) = c$

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$\frac{d}{d x} (u \pm v) = u^{'} \pm v^{'}$

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$\frac{d}{d x} (u \cdot v) = u v^{'} + u^{'} v$

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$\frac{d}{d x} (\frac{u}{v}) = \frac{v u^{'} - u v^{'}}{v^{2}}$

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$\frac{d}{d x} (u (v)) = u^{'} (v) v^{'}$

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$\frac{d}{d x} (c) = 0$

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$\frac{d}{d x} (x) = 1$

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List B.1.2. Derivatives of Elementary Functions

$\frac{d}{d x} (x^{n}) = n x^{n - 1}$

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$\frac{d}{d x} (e^{x}) = e^{x}$

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$\frac{d}{d x} (a^{x}) = \ln a \cdot a^{x}$

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$\frac{d}{d x} (\ln x) = \frac{1}{x}$

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$\frac{d}{d x} (\log_{a} x) = \frac{1}{\ln a} \cdot \frac{1}{x}$

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$\frac{d}{d x} (\sin (x)) = \cos (x)$

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$\frac{d}{d x} (\cos (x)) = - \sin (x)$

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$\frac{d}{d x} (\csc (x)) = - \csc (x) \cot (x)$

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$\frac{d}{d x} (\sec (x)) = \sec (x) \tan (x)$

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$\frac{d}{d x} (\tan (x)) = \sec^{2} (x)$

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$\frac{d}{d x} (\cot (x)) = - \csc^{2} (x)$

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$\frac{d}{d x} (\cosh = (x)) = \sinh (x)$

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$\frac{d}{d x} (\sinh (x)) = \cosh (x)$

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$\frac{d}{d x} (sech (x)) = - sech (x) \tanh (x)$

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$\frac{d}{d x} (\tanh (x)) = {sech}^{2} (x)$

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$\frac{d}{d x} (csch (x)) = - csch (x) \coth (x)$

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$\frac{d}{d x} (\coth (x)) = - {csch}^{2} (x)$

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List B.1.3. Derivatives of Inverse Functions

$\frac{d}{d x} (\sin^{- 1} (x)) = \frac{1}{\sqrt{1 - x^{2}}}$

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$\frac{d}{d x} (\cos^{- 1} (x)) = \frac{- 1}{\sqrt{1 - x^{2}}}$

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$\frac{d}{d x} (\csc^{- 1} (x)) = \frac{- 1}{| x | \sqrt{x^{2} - 1}}$

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$\frac{d}{d x} (\sec^{- 1} (x)) = \frac{1}{| x | \sqrt{x^{2} - 1}}$

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$\frac{d}{d x} (\tan^{- 1} (x)) = \frac{1}{1 + x^{2}}$

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$\frac{d}{d x} (\cot^{- 1} (x)) = \frac{- 1}{1 + x^{2}}$

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$\frac{d}{d x} (\cosh^{- 1} (x)) = \frac{1}{\sqrt{x^{2} - 1}}$

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$\frac{d}{d x} (\sinh^{- 1} (x)) = \frac{1}{\sqrt{x^{2} + 1}}$

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$\frac{d}{d x} ({sech}^{- 1} (x)) = \frac{- 1}{x \sqrt{1 - x^{2}}}$

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$\frac{d}{d x} ({csch}^{- 1} (x)) = \frac{- 1}{| x | \sqrt{1 + x^{2}}}$

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$\frac{d}{d x} (\tanh^{- 1} (x)) = \frac{1}{1 - x^{2}}$

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$\frac{d}{d x} (\coth^{- 1} (x)) = \frac{1}{1 - x^{2}}$

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