Handout on calculus. Total derivatives d/dx[F(x)+G(x)l=d/dx[F(x)+d/dxG(x) example: 3x+5x= dr(3x2)+d (5x3)=6x+15x d/dx[F(x).G(x)1= F(x)d/dx[G(x)+G(x)d/dx[F(x)] example. d 13*3x (3x2)+3 (5x1)=5x*6x+3,5 d/dx[F(x)/G(x)=[G(x)d/dx[F(x)l-F(x)d I dx[G(x) 5x1/3d ④+(3x2 )-3x2(5x3) example: [3x2/5x= d/dx[F(g(x)= dF/dG. d/dxG(x)) d d 6 In the last function which is the case of function of a function, identify there are two functions in(3x2)3. The F function is the cube root function and G function is the 3 In this simple case indeed we could have made it one function which is 3x. Consider the more complex case of the following function of a function case To find the derivative of In(-). here the first function is the log function and the x+2 second function is(-). Hence the derivative is given as 1(x+2)*3-3x x+2 (x+2)2 2 deriving the derivatives in the examples above, derivatives of some common functional forms have been used which are given below.Handout on Calculus: Total derivatives 2 1 / 2 2 1 / 2 (1 / 2) 1 2 1 / 3 2 1 / 3 2 2 1 / 3 2 1 / 3 2 1 / 3 2 1 / 3 2 2 1 / 3 1 / 3 2 2 / 3 2 3 2 3 2 2 * (3 ) 6 (3 ) (3 ) 2 1 : [3 ] / [ ( ( )] / .{ / [ ( )}] (5 ) 5 (3 ) 3 (5 ) : [3 / 5 ] { ( )} [ ( ) / [ ( )] ( ) / [ ( )]] / [ ( ) / ( )] 3 5 : [5 * 3 ] 5 (3 ) 3 (5 ) 5 * 6 3 * / [ ( ). ( )] ( ) / [ ( )] ( ) / [ ( )] : [3 5 ] (3 ) (5 ) 6 15 / [ ( ) ( )] / [ ( )] / [ ( )] x x x dx d x x dx d example d dx F G x dF dG d dx G x x x dx d x x dx d x x x dx d example G x G x d dx F x F x d dx G x d dx F x G x x x x x x dx d x x dx d x x x dx d example d dx F x G x F x d dx G x G x d dx F x x x x dx d x dx d x x dx d example d dx F x G x d dx F x d dx G x = = = − = − = = + = + = + + = + = + + = + − − In the last function which is the case of function of a function, identify there are two functions in 2 1/ 3 (3x ) . The F function is the cube root function and G function is the 2 3x . In this simple case indeed we could have made it one function which is 2 / 3 3x . Consider the more complex case of the following function of a function case. To find the derivative of ) 2 3 ln( x + x . Here the first function is the Log function and the second function is ) 2 3 ( x + x . Hence the derivative is given as: 2 ( 2) ( 2) *3 3 2 3 1 + + − + x x x x x . In deriving the derivatives in the examples above, derivatives of some common functional forms have been used which are given below