# Differential calculus - Table of derivatives

Independent variable : x
Real constants : C,a,b,c
Natural number : n

1. ${d \over dx}(C)=0$
2. ${d \over dx}(x)=1$
3. ${d \over dx}(ax+b)=a$
4. ${d \over dx}(ax^2+bx+c)=ax+b$
5. ${d \over dx}(x^n)=nx^{n-1}$
6. ${d \over dx}(x^{-n})=-{n \over x^{n+1} }$
7. ${d \over dx} ({1 \over x })= -{1 \over x^2 }$
8. ${d \over dx}(\sqrt x)={1 \over 2 \sqrt x }$
9. ${d \over dx}(\sqrt [n] x)={1 \over n \sqrt [n] {x^{n-1} }}$
10. ${d\over dx} (In\,x)= - {1 \over x}$
11. ${d \over dx} (log_ax)= {1 \over x\, In \,a }, a> 0 ,a \neq 1.$
12. ${d \over dx } (a^x)= a^xIn\,a\,, a>0, a\neq1.$
13. ${d \over dx } (e^x)=e^x$
14. ${d \over dx }(sin\,x)= cos \, x$
15. ${d \over dx }(cos\,x)= -sin \, x$
16. ${d \over dx }(tan\,x)= {1 \over cos^2 x }= sec^2x$
17. ${d \over dx }(cot\,x)= -{1 \over sin^2 x }= -csc^2x$
18. ${d \over dx }(sec\,x )=tan\,x.sec\,x$
19. ${d \over dx }(csc\,x )=-cot\,x.csc\,x$
20. ${d \over dx }(arcsin\,x)= {1 \over \sqrt {1-x^2}}$
21. ${d \over dx }(arccos\,x)=-{1 \over \sqrt {1-x^2}}$
22. ${d \over dx }(arctan\,x)={1 \over 1+x^2}$
23. ${d \over dx }(arccot\,x)=-{1 \over 1+x^2}$
24. ${d \over dx }(arcsec\,x)= {1 \over |x|\sqrt {x^2-1}}$
25. ${d \over dx }(arccsc\,x)= -{1 \over |x|\sqrt {x^2-1}}$
26. ${d \over dx} (sinh \, x)=cosh \, x$
27. ${d \over dx} (cosh \, x)=sinh \, x$
28. ${d \over dx} (tanh \, x)= {1 \over cosh^2x }= sech^2x$
29. ${d \over dx} (coth \, x)= -{1 \over sinh^2x }= -csch^2x$
30. ${d \over dx} (sech \, x)= -sech \,x. tanh\, x$
31. ${d \over dx} (csch \, x)= -csch \,x. coth\, x$
32. ${d \over dx } (arcsinh \, x)= {1 \over \sqrt{x^2+1}}$
33. ${d \over dx } (arccosh \, x)= {1 \over \sqrt{x^2-1}}$
34. ${d \over dx } (arctanh \, x)= {1 \over 1-x^2}, |x| < 1.$
35. ${d \over dx } (arccoth \, x)= -{1 \over x^2-1}, |x| >1.$
36. ${d \over dx } (u^v)=vu^{v-1}.{du \over dx }+u^vIn\, u.{dv \over dx}$