$$\begin{align*} F(x) & \ - \textrm{ diff. } \to \quad F'(x) \nl \int f(x)\;dx & \ \ \leftarrow \textrm{ int. } - \quad f(x) \nl a &\qquad\qquad\qquad 0 \nl x &\qquad\qquad\qquad 1 \nl af(x) &\qquad\qquad\qquad af'(x) \nl f(x)+g(x) &\qquad\qquad\qquad f'(x)+g'(x) \nl x^n &\qquad\qquad\qquad nx^{n-1} \nl 1/x=x^{-1} &\qquad\qquad\qquad -x^{-2} \nl \sqrt{x}=x^{\frac{1}{2}} &\qquad\qquad\qquad \frac{1}{2}x^{-\frac{1}{2}} \nl {\rm e}^x &\qquad\qquad\qquad {\rm e}^x \nl a^x &\qquad\qquad\qquad a^x\ln(a) \nl \ln(x) &\qquad\qquad\qquad 1/x \nl \log_a(x) &\qquad\qquad\qquad (x\ln(a))^{-1} \nl \sin(x) &\qquad\qquad\qquad \cos(x) \nl \cos(x) &\qquad\qquad\qquad -\sin(x) \nl \tan(x) &\qquad\qquad\qquad \sec^2(x)\equiv\cos^{-2}(x) \nl \csc(x) \equiv \frac{1}{\sin(x)} &\qquad\qquad\qquad -\sin^{-2}(x)\cos(x) \nl \sec(x) \equiv \frac{1}{\cos(x)} &\qquad\qquad\qquad \tan(x)\sec(x) \nl \cot(x) \equiv \frac{1}{\tan(x)} &\qquad\qquad\qquad -\csc^2(x) \nl \sinh(x) &\qquad\qquad\qquad \cosh(x) \nl \cosh(x) &\qquad\qquad\qquad \sinh(x) \nl \sin^{-1}(x) &\qquad\qquad\qquad \frac{1}{\sqrt{1-x^2}} \nl \cos^{-1}(x) &\qquad\qquad\qquad \frac{-1}{\sqrt{1-x^2}} \nl \tan^{-1}(x) &\qquad\qquad\qquad \frac{1}{1+x^2} \end{align*}$$