DERIVATION OF DIFFERENTIATION(WITH EXAMPLES)

                                                                 DIFFERENTIATION


 THE TRIANGLE FORMED ON TOP IS 










             tanθ= dy/dx..............from above figure

             as slope is tanθ is slope(say m);

             so  m= dy/dx.

             as dy = F(X+dx) - F(X);      FROM FIRST FIGURE.

             SO     m= F(X+dx)-F(X)/dx

              now, dx is very small;

              let dx=h;

            dy/dx=   lim   F(X+h)-F(X)   /   h  ....................1.

                         h->0

                                                          SOME EXAMPLES

 1. F(X)=Y=SINX.

    FIND dy/dx?

ANS.

    F(X)= SINX  =>  F(X+h) = SIN(X+h)

   AS   dy/dx =  lim F(X+h)-F(X)  /  h  ....................from 1.

                        h->0

     dy/dx  = lim  SIN(X+h)- SIN(X)  / h

                    h->0


      dy/dx =lim  (SINX COSh + COSX SINh) - SINX     /     h

                   h->0

      dy/dx = lim     (SINX COSh + COSX SINh - SINX)  / h

                   h->0

as h is very close to 0 ;   COSh = 1  and SINh=h;

      dy/dx = lim   ( SINX + COSX . h  -  SINX)   / h

                  h->0

      dy/dx = lim( COSX. h)  / h

                  h->0

      dy/dx= COSX


                 SO DIFFERENTIATION OF SINX IS COSX.

 

2. Y=4X

FIND dy/dx=?

ANS

 F(X)=4X     SO     F(X+h)=4(X+h)

dy/dx= lim F(X+h)-F(X)   /   h

            h->0

         = lim 4(X+h)-4X   /    h

           h->0

         =lim  (4X+4h-4X)  / h

          h->0

        =lim  4h /h

        =4.


SO DIFFERENTIATION OF 4x IS  4.

 





           


Comments

Post a Comment

Popular posts from this blog

DERIVATION OF AREA AND CIRCUMFERENCE OF CIRCLE USING CALCULUS

Image
CIRCLE DERIVATION OF AREA OF CIRCLE LETS TAKE d𝞡 , A  VERY SMALL ANGLE IN ABOVE FIGURE SO; THE SMALL PART OF CIRCLE IN DIAGRAM  CAN BE CONSIDERED AS RIGHT ANGLED TRIANGLE FROM ABOVE FIGURE, tan d𝞡= opposite side/ adjacent side                                                     tan d𝞡   = ds/r AS ANGLE IS VERY SMALL, tan d𝞡 = d𝞡  SO,                                               d𝞡 =ds/r SO,                                                ds=r d𝞡 ..................(1)                                                      AREA OF TRIANGLE= 1/2 * BASE *HEIGHT                                       = 1/2 * r * ds                                    D(A)   = 1/2 * r *r d𝞡.............      FROM   (1) IF WE INTEGRATE THESE SMALL TRIANGLES HAVING THIS AREA, WE WILL GET AREA OF CIRCLE                                                SO     AREA OF CIRCLE= ∫D(A)                                                            2𝜋                                                       =  ∫    1/2* r
Read more

L'HOSPITAL'S RULE (WITH EXAMPLES)

                               L'HOSPITAL'S                                                                 RULE IF  Lim  F(X)/G(X) = 0/0 ;       X->a THEN   Lim F(X)/G(X)  =  Lim F'(X)/G'(X)                X->a                         X->a PROBLEMS BASED ON ABOVE RULE:                                   _ 1. Lim   sin(1-x)/√x - 1 = ?    x->1 ANSWER: IF WE PUT x=1; WE GET  ...........  sin(1-x)                                                                = sin(1-1)                                                                 =sin(0)                                                 _              =0                AND .................. √x - 1                                              =1-1                                               =0 SO,                        _           sin(1-x)/√x - 1 = 0/o SO APPLY L'HOSPITAL'S RULE;                               _                                                                  _  Lim   sin(1-x)/√x - 1
Read more