Here we will list the derivatives of several basic functions and their derivation processes: 1.y = c (where c is a constant) y' = 02. y = xny' = NX (n-1) 3. y = axy' = axlnay = exy'. x5.y=sinxy'=cosx6.y=cosxy'=-sinx7.y=tanxy'= 1/cos^2x8.y=cotxy'=- 1/sin^2x9.y=arcsinxy'= 1/√ 1-x^2 1 0.y=arccosxy'=- 1/√ 1-x 2 1 1 . y = arctan xy ' = 1/ 1x 2 12 . y = arccotxy ' =- 1/ 1。 In y' = f' [g(x)] g'(x)' f' [g (x)], g(x) is regarded as a whole variable, while in g'(x), x is regarded as the variable "2. y = u/v，y' = u' v-uv'/v 23。 The definition of derivative is the same: y=c,⊿y=c-c=0,lim⊿x→0⊿y/⊿x=0.2. This derivation is not proved for the time being, because if it is deduced according to the definition of derivative, it cannot be extended to the general case that n is an arbitrary real number. Two results, y = exy' = exx and y=lnxy'= 1/x, can be proved by the derivative of composite function. 3.y = a x, ⊿ y = a (x ⊿ x)-a x = a x (a ⊿ x-1) ⊿ y/⊿ x = a x (a we can know from the auxiliary function: ⊿ x = So (a ⊿ x-1)/⊿ x = β/loga (1β) =1β/loga (1β)1β Obviously, when ⊿. Substituting this result into lim⊿x→0⊿y/ ⊿ x = lim ⊿ x → 0ax (a ⊿ x-1)/⊿ x gives lim ⊿ x → 0 ⊿ y/. It can be known that when a=e, there is y = exy' = exx.4. Y = logax ⊿ y = loga (x ⊿ x)-logax = loga (x ⊿ x)/x = loga [(1⊿ x/x) x]/x ⊿ y/⊿. Because y = x n and y = e ln (x n) = e NLNX, y' = e nlnx (nlnx)' = x nn/x = nx (n-1). 5.y = sinx⊿y = sin(x⊿x)-sinx = 2cos(x⊿x/2)sin(⊿x/2)⊿y/⊿x = 2cos(x⊿x/)7 . y = tanx = sinx/cosx y'=[(sinx)'cosx-sinx(cos)']/cos^2x=(cos^2x sin^2x)/ Cos2x =1/cos 2x8.y = cotx = cosx/sinxy' = [(cosx)' sinx-cosx (sinx)']/sin2x =-1/sin2x9.y = arcsinx = There are many.