Differentiation Rules

General Formulas

Assume \(u\) and \(v\) are differentiable functions of \(x\), \(c\) is a constant.

\[u'=\diff{u}{x}, \quad v'=\diff{v}{x}\]

1. \(\displaystyle \diff{c}{x}=0\)

2. \(\displaystyle \diff{}{x}(u+v)=u' + v'\)

3. \(\displaystyle \diff{}{x}(u-v)=u' - v'\)

4. \(\displaystyle \diff{}{x}(cu) = c u'\)

5. \(\displaystyle \diff{}{x}(uv) = u'v + u v' \)

6. \(\displaystyle \diff{}{x}\biggl(\frac{u}{v}\biggr) = \frac{u'v - u v'}{v^2} \)

7. \(\displaystyle \diff{}{x} (x^n) = n x^{n-1}\)

8. \(\displaystyle \diff{}{x}\biggl\{f\biggl(g(x)\biggr)\biggr\}= \diff{f}{g} \cdot \diff{g}{x}\)

Trigonometric Functions

9. \(\displaystyle \diff{}{x}(\sin x) = \cos x\)

10. \(\displaystyle \diff{}{x}(\cos x) = -\sin x\)

11. \(\displaystyle \diff{}{x}(\tan x) = \sec^2 x\)

12. \(\displaystyle \diff{}{x}(\sec x) = \sec x \tan x\)

13. \(\displaystyle \diff{}{x}(\cot x) = -\csc^2 x\)

14. \(\displaystyle \diff{}{x}(\csc x) = -\csc x \cot x\)

Inverse Trigonometric Functions

15. \(\displaystyle \diff{}{x}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}\)

16. \(\displaystyle \diff{}{x}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}\)

17. \(\displaystyle \diff{}{x}(\arctan x) = \frac{1}{1+x^2}\)

18. \(\displaystyle \diff{}{x}(\arcsec x) = \frac{1}{|x|\sqrt{x^2-1}}\)

19. \(\displaystyle \diff{}{x}(\arccot x) = -\frac{1}{1+x^2}\)

20. \(\displaystyle \diff{}{x}(\arccsc x) = -\frac{1}{|x|\sqrt{x^2-1}}\)

Exponential and Logarithmic Functions

21. \(\displaystyle \diff{}{x} (e^x) = e^x\)

22. \(\displaystyle \diff{}{x} (a^x) = a^x \ln a\)

23. \(\displaystyle \diff{}{x} (\ln x) = \frac{1}{x}\)

24. \(\displaystyle \diff{}{x} (\log_a x) = \frac{1}{x \ln a}\)

Hyperbolic Functions

25. \(\displaystyle \diff{}{x}(\sinh x) = \cosh x\)

26. \(\displaystyle \diff{}{x}(\cosh x) = \sinh x\)

27. \(\displaystyle \diff{}{x}(\tanh x) = \sech^2 ~x\)

28. \(\displaystyle \diff{}{x}(\sech x) = -\sech ~x \tanh x\)

29. \(\displaystyle \diff{}{x}(\coth x) = -\csch^2 ~x\)

30. \(\displaystyle \diff{}{x}(\csch x) = -\csch ~x ~\coth x\)

Inverse Hyperbolic Functions

Parametric Equations