Introduction : Differentiation formulas of all types of mathematical functions are given below

Differentiation of algebraic functions

  1. \(\frac{d}{dx}(c)=0\) where c is any constant.
  2. \(\frac{d}{dx}(1)=0\)
  3. \(\frac{d}{dx}(x)=1\)
  4. \(\frac{d}{dx}(cx)=c\) where c is a costant.
  5. \(\frac{d}{dx}(ax+b=a\) where a , b are constants.
  6. \(\frac{d}{dx}(x^r)=rx^{r-1}\) for any number r.
  7. \(\frac{d}{dx}(\sqrt{x})=\frac{1}{2\sqrt{x}}\)
  8. \(\frac{d}{dx}(\frac{1}{\sqrt{x}})=-\frac{1}{2x\sqrt{x}}\)

Differentiation of logarithmic functions

  1. \(\frac{d}{dx}(ln(x))=\frac{1}{x}\) differentiation natural logarithm
  2. \(\frac{d}{dx}(ln(ax+b))=\frac{a}{ax+b}\)
  3. \(\frac{d}{dx}(log_a(x))=\frac{log_e(a)}{x}\) differentiation logarithm with base a.
  4. \(\frac{d}{dx}(log_a(cx+d))=c\frac{log_e(a)}{cx+d}\)

Differentiation of exponential functions

  1. \(\frac{d}{dx}(e^x)=e^x\)
  2. \(\frac{d}{dx}(e^{-x})=-e^{-x}\)
  3. \(\frac{d}{dx}(a^x)=a^x log_e(a)\)
  4. \(\frac{d}{dx}(a^{-x})=-a^{-x} log_e(a)\)

Differentiation of trigonometric functions

  1. \(\frac{d}{dx}(\sin (x))=\cos (x)\)
  2. \(\frac{d}{dx}(\cos (x))=\sin (x)\)
  3. \(\frac{d}{dx}(\tan(x))=\sec ^2(x)\)
  4. \(\frac{d}{dx}(\csc(x))=-\cot (x) \csc(x)\)
  5. \(\frac{d}{dx}(\sec(x))=\tan (x) \sec (x)\)
  6. \(\frac{d}{dx}(\cot(x))=-\csc^2(x)\)

Differentiation of inverse trigonometric functions

  1. \(\frac{d}{dx}(\sin^{-1} (x))=\frac{1}{\sqrt{1-x^2}}\)
  2. \(\frac{d}{dx}(\cos^{-1} (x))=-\frac{1}{\sqrt{1-x^2}}\)
  3. \(\frac{d}{dx}(\tan^{-1}(x))=\frac{1}{x^2+1}\)
  4. \(\frac{d}{dx}(\csc^{-1}(x))=-\frac{1}{\sqrt{1-\frac{1}{x^2}} x^2}\)
  5. \(\frac{d}{dx}(\sec^{-1}(x))=\frac{1}{\sqrt{1-\frac{1}{x^2}} x^2}\)
  6. \(\frac{d}{dx}(\cot^{-1}(x))=-\frac{1}{x^2+1}\)

Differentiation of trigonometric hyperbolic functions

  1. \(\frac{d}{dx}(\sinh (x))=\cosh (x)\)
  2. \(\frac{d}{dx}(\cosh (x))=\sinh (x)\)
  3. \(\frac{d}{dx}(\tanh(x))=\text{sech}^2(x)\)
  4. \(\frac{d}{dx}(\text{csch})=-\coth (x) \text{csch}(x)\)
  5. \(\frac{d}{dx}(\text{sech}(x))=\tanh (x) (-\text{sech}(x))\)
  6. \(\frac{d}{dx}(\coth(x))=-\text{csch}^2(x)\)

Differentiation of trigonometric inverse hyperbolic functions

  1. \(\frac{d}{dx}(\sinh^{-1} (x))=\frac{1}{\sqrt{x^2+1}}\)
  2. \(\frac{d}{dx}(\cosh^{-1} (x))=1/(Sqrt[-1 + x] Sqrt[1 + x])\)
  3. \(\frac{d}{dx}(\tanh^{-1}(x))=\frac{1}{x^2+1}\)
  4. \(\frac{d}{dx}(\text{csch}^{-1}(x))=-(1/(Sqrt[1 + 1/x^2] x^2))\)
  5. \(\frac{d}{dx}(\text{sech}^{-1}(x))=-\frac{1}{x \sqrt{\frac{1-x}{x+1}} (x+1)}\)
  6. \(\frac{d}{dx}(\coth^{-1}(x))=\frac{1}{1-x^2}\)