### Math Notes

Subjects

#### Differential Calculus Solutions

##### Topics || Problems

Let u and v be functions of x: a, n and c be constants.

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. $$\frac{d}{{dx}}\left( {{\mathop{\rm csch}\nolimits} u} \right) = - \left( {{\mathop{\rm csch}\nolimits} u} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)$$

Identities of Hyperbolic Functions

$$\sinh y = \frac{{{e^y} - {e^{ - y}}}}{2}$$

$$\cosh y = \frac{{{e^y} + {e^{ - y}}}}{2}$$

$$\tanh y = \frac{{\sinh y}}{{\cosh y}} = \frac{{{e^y} - {e^{ - y}}}}{{{e^y} + {e^{ - y}}}}$$

$$\coth y = \frac{{\cosh y}}{{\sinh y}} = \frac{{{e^y} + {e^{ - y}}}}{{{e^y} - {e^{ - y}}}}$$

$${\mathop{\rm sech}\nolimits} y = \frac{1}{{\cosh y}} = \frac{2}{{{e^y} + {e^{ - y}}}}$$

$${\mathop{\rm csch}\nolimits} y = \frac{1}{{\sinh y}} = \frac{2}{{{e^y} - {e^{ - y}}}}$$

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. $$\frac{d}{{dx}}\left( {arc{\mathop{\rm csch}\nolimits} u} \right) = - \left( {\frac{1}{{u\sqrt {{u^2} + 1} }}} \right)\left( {\frac{{d\left( u \right)}}{{dx}}} \right)$$