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$$\begin{array}{cc} f&f'\\ \hline \alpha f+\beta g&\alpha f'+\beta g'\\ x^\alpha&\alpha x^{\alpha-1}\\ \exp&\exp\\ \sin&\cos\\ \cos&-\sin\\ \sinh&\cosh\\ \cosh&\sinh\\ \ln&\frac{1}{x}\\ fg&f'g+fg'\\ \frac{f}{g}&\frac{f'g-fg'}{g^2}\\ f(g(x))&f'(g(x))g'(x)\\ \end{array}$$
Paul's Derivatives/Integrals