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Differential calculus - Definition and properties of the derivative

Functions : f,g,y,u,v

Independent variable : x

Real constant : k

Angle :\(\alpha\)

  1. \(y'(x)= \displaystyle \lim_{\bigtriangleup x \to 0} {\bigtriangleup y \over \bigtriangleup x} ={dy \over dx }\)
  2. \({dy \over dx }= tan \alpha\)
  3. \({d(u+v) \over dx} ={du \over dx}+{dv \over dx }\)
  4. \({d(u-v) \over dx} ={du \over dx}-{dv \over dx }\)
  5. \({d(ku) \over dx} = k {du \over dx}\)
  6. Product Rule 
    \({d(u.v) \over dx }= {du \over dx }.v+u.{dv \over dx }\)
  7. Quotient Rule
    \({d \over dx} \bigg({u \over v}\bigg)={{du \over dx}.v-u.{dv \over dx} \over v^2}\)
  8. Chain Rule 
    \(y=f(g(x)),u=g(x),\)
    \({dy \over dx}={dy \over du}.{du \over dx}\)
  9. Derivative of Inverse Function 
    \({ dy \over dx } = {1 \over {dx \over dy}},\)
    Where x(y) is the inverse function of y(x).
  10. Reciprocal Rule 
    \({d \over dx } \bigg ( {1 \over y }\bigg) =-{{dy \over dx} \over y^2}\)
  11. Logarithmic Differentiation 
    \(y=f(x),In \; y= In \;f(x),\)
    \({dy \over dx } =f(x). {d \over dx}[In\; f(x)].\)