Some Calculus Formulas

Derivative

Polynomial

1. $ c’ = 0 $
2. $ x’ = 1 $
3. $ (cx)’ = c $
4. $ |x|’ = \frac{x}{|x|}, x \ne 0 $
5. $ (x^c)’ = cx^{c-1} $
6. $ \left(\frac 1 x\right)’ = (x^{-1})’ = -x^{-2} = - \frac {1} {x^2} $
7. $ \left(\frac 1 {x^c}\right)’ = (x^{-c})’ = -cx^{-c-1} = - \frac {1} {x^{c+1}} $
8. $ \left(\sqrt{x}\right)’ = \left(\frac 1 {x^{1/2}}\right)’ = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}, x \gt 0 $

Exponential / Logarithmic

1. $ (c^x)’ = c^x \ln{c}, c \gt 0 $
2. $ (e^x)’ = e^x $
3. $ (\log_cx)’ = \frac{1}{x\ln{c}}, c \gt0, c \ne 1 $
4. $ (\ln{x})’ = \frac{1}{x}, x \gt 0 $
5. $ (\ln{|x|})’ = \frac{1}{x} $
6. $ (x^x)’ = x^x(1 + \ln{x}) $

Trigonometric

1. $ (\sin{x})’ = \cos{x} $
2. $ (\cos{x})’ = -\sin{x} $
3. $ (\tan{x})’ = \sec^2{x} = \frac{1}{\cos^2x} $
4. $ (\sec{x})’ = \sec{x} * \tan{x} $
5. $ (\csc{x})’ = -\csc{x} * \cot{x} $
6. $ (\cot{x})’ = -\csc^2{x} = \frac{-1}{\sin^2{x}} $
7. $ (\arcsin{x})’ = \frac{1}{\sqrt{1 - x^2}} $
8. $ (\arccos{x})’ = \frac{-1}{\sqrt{1 - x^2}} $
9. $ (\arctan{x})’ = \frac{1}{1 + x^2} $
10. $ (arcsec \hskip1pt x)’ = \frac{1}{|x|\sqrt{x^2 - 1}} $
11. $ (arccsc \hskip1pt x)’ = \frac{-1}{|x|\sqrt{x^2 - 1}} $
12. $ (arccot \hskip1pt x)’ = \frac{-1}{1 + x^2} $

Hyperbolic

1. $ (\sinh{x})’ = \cosh{x} = \frac{e^x + e^{-x}}{2} $
2. $ (\cosh{x})’ = \sinh{x} = \frac{e^x - e^{-x}}{2} $
3. $ (\tanh{x})’ = sech^2 \hskip1pt {x} $
4. $ (sech \hskip1pt {x})’ = -\tanh{x} * sech \hskip1pt {x} $
5. $ (csch \hskip1pt {x})’ = -\coth{x} * csch \hskip1pt {x} $
6. $ (\coth{x})’ = - csch^2 \hskip1pt {x} $
7. $ (arcsinh \hskip1pt {x})’ = \frac{1}{\sqrt{x^2 + 1}} $
8. $ (arccosh \hskip1pt {x})’ = \frac{1}{\sqrt{x^2 - 1}}, x \gt 1 $
9. $ (arctanh \hskip1pt {x})’ = \frac{1}{1 - |x|^2}, |x| \lt 1 $
10. $ (arsech \hskip1pt {x})’ = \frac{-1}{x\sqrt{1 - x^2}} $
11. $ (arcsch \hskip1pt {x})’ = \frac{-1}{x\sqrt{1 + x^2}} $
12. $ (arcoth \hskip1pt {x})’ = \frac{1}{1 - |x|^2}, |x| \gt 1 $