# 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
$y=f(x),In \; y= In \;f(x),$
${dy \over dx } =f(x). {d \over dx}[In\; f(x)].$