Algebraic Functions
1. Derivative of a constant: \(\frac{d}{{dx}}\left( c \right) = 0\)
2. Power Formula:\(\frac{d}{{dx}}\left( {{u^n}} \right) = \frac{d}{{dx}}\left( {{u^n}} \right) = \left( n \right)\left( {{u^{n - 1}}} \right)\frac{{du}}{{dx}} = n\left( {{u^{n - 1}}\frac{{du}}{{dx}}} \right)\)
3. Derivative of a product:\(\frac{d}{{dx}}\left( {uv} \right) = v\frac{d}{{dx}}\left( u \right) + u\frac{d}{{dx}}\left( v \right)\)
4. Derivative of a quotient:\(\frac{d}{{dx}}\left( {\frac{u}{v}} \right) = \frac{{v\frac{d}{{dx}}\left( u \right) - u\frac{d}{{dx}}(v)}}{{{v^2}}}\)
5. Derivative of sum or difference:\(\frac{d}{{dx}}\left( {u \pm v} \right) = \frac{d}{{dx}}\left( u \right) \pm \frac{d}{{dx}}\left( v \right)\)
Exponential and Logarithmic
7. \(\frac{d}{{dx}}{\mathop{\rm (lnu)}\nolimits} = \frac{{\frac{d}{{dx}}\left( u \right)}}{u}\)
8. \(\frac{d}{{dx}}\left( {{{\log }_a}u} \right) = \frac{{\frac{d}{{dx}}\left( u \right)}}{{u\ln a}}\)
9. \(\frac{d}{{dx}}\left( {{a^u}} \right) = {a^u}\ln a\frac{d}{{dx}}\left( u \right)\)
10. \(\frac{d}{{dx}}\left( {{e^u}} \right) = {e^u}\frac{d}{{dx}}\left( u \right)\)
11. \(\frac{d}{{dx}}\left( {{u^v}} \right) = u{v^{v - 1}}\left( {\frac{{d\left( u \right)}}{{dx}}} \right) + {u^v}\left( {\ln u} \right)\frac{{d\left( v \right)}}{{dx}}\)
Trigonometric
12. \(\frac{d}{{dx}}\left( {{\mathop{\rm sinu}\nolimits} } \right) = {\mathop{\rm cosu}\nolimits} \frac{{d\left( u \right)}}{{dx}}\)
13. \(\frac{d}{{dx}}\left( {\cos u} \right) = - \sin u\frac{{d\left( u \right)}}{{dx}}\)
14. \(\frac{d}{{dx}}\left( {\tan u} \right) = {\sec ^2}u\frac{{d\left( u \right)}}{{dx}}\)
15. \(\frac{d}{{dx}}\left( {\sec u} \right) = \sec u\tan u\frac{{d\left( u \right)}}{{dx}}\)
16. \(\frac{d}{{dx}}\left( {\cot u} \right) = - {\csc ^2}u\frac{{d\left( u \right)}}{{dx}}\)
17. \(\frac{d}{{dx}}\left( {\csc u} \right) = - \csc u\cot u\frac{{d\left( u \right)}}{{dx}}\)
Inverse Trigonometric Functions
18. \(\frac{d}{{dx}}\left( {\arcsin u} \right) = \left( {\frac{1}{{\sqrt {1 - {u^2}} }}} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
19. \(\frac{d}{{dx}}\left( {\arccos u} \right) = - \left( {\frac{1}{{\sqrt {1 - {u^2}} }}} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
20. \(\frac{d}{{dx}}\left( {\arctan u} \right) = \left( {\frac{1}{{1 + {u^2}}}} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
21. \(\frac{d}{{dx}}\left( {arc\cot } \right) = - \left( {\frac{1}{{1 + {u^2}}}} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
22. \(\frac{d}{{dx}}\left( {arc\sec u} \right) = \left( {\frac{1}{{u\sqrt {{u^2} - 1} }}} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
23. \(\frac{d}{{dx}}\left( {arc\csc u} \right) = - \left( {\frac{1}{{u\sqrt {{u^2} - 1} }}} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
Hyperbolic Functions
24. \(\frac{d}{{dx}}\left( {\sinh u} \right) = \left( {\cosh u} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
25. \(\frac{d}{{dx}}\left( {\cosh u} \right) = \left( {\sinh u} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
26. \(\frac{d}{{dx}}\left( {\tanh u} \right) = \left( {{{\sec }^2}u} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
27. \(\frac{d}{{dx}}\left( {\coth u} \right) = - \left( {\csc {h^2}u} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
28. \(\frac{d}{{dx}}\left( {{\mathop{\rm sech}\nolimits} u} \right) = - \left( {{\mathop{\rm sech}\nolimits} u\tanh u} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
29. \(\frac{d}{{dx}}\left( {{\mathop{\rm csch}\nolimits} u} \right) = - \left( {{\mathop{\rm csch}\nolimits} u} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
\(\sinh y = \frac{{{e^y} - {e^{ - y}}}}{2}\)
\(\cosh y = \frac{{{e^y} + {e^{ - y}}}}{2}\)
\(\tanh y = \frac{{\sinh y}}{{\cosh y}} = \frac{{{e^y} - {e^{ - y}}}}{{{e^y} + {e^{ - y}}}}\)
\(\coth y = \frac{{\cosh y}}{{\sinh y}} = \frac{{{e^y} + {e^{ - y}}}}{{{e^y} - {e^{ - y}}}}\)
\({\mathop{\rm sech}\nolimits} y = \frac{1}{{\cosh y}} = \frac{2}{{{e^y} + {e^{ - y}}}}\)
\({\mathop{\rm csch}\nolimits} y = \frac{1}{{\sinh y}} = \frac{2}{{{e^y} - {e^{ - y}}}}\)
Inverse Hyperbolic Functions
30. \(\frac{d}{{dx}}\left( {{\mathop{\rm arcsinh}\nolimits} u} \right) = \left( {\frac{1}{{\sqrt {{u^2} + 1} }}} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
31. \(\frac{d}{{dx}}\left( {{\mathop{\rm arccosh}\nolimits} u} \right) = \left( {\frac{1}{{\sqrt {{u^2} - 1} }}} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
32. \(\frac{d}{{dx}}\left( {{\mathop{\rm arctanh}\nolimits} u} \right) = \left( {\frac{1}{{1 - {u^2}}}} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
33. \(\frac{d}{{dx}}\left( {arc\coth u} \right) = - \left( {\frac{1}{{{u^2} - 1}}} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
34. \(\frac{d}{{dx}}\left( {arc{\mathop{\rm sech}\nolimits} u} \right) = - \left( {\frac{1}{{u\sqrt {1 - {u^2}} }}} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)
35. \(\frac{d}{{dx}}\left( {arc{\mathop{\rm csch}\nolimits} u} \right) = - \left( {\frac{1}{{u\sqrt {{u^2} + 1} }}} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)\)