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Consider the function f(x) = sin -1 (3x-4x 3 ) + cos -1 (4x 3 -3x) Let the dreivative of the function be, f'(x) = D 1 +D 2 where D 1 = (d/dx)(sin -1 (3x-4x 3 )) -------------(1) and D 2 = (d/dx)(cos -1 (4x 3 -3x)) ------------(2) Let x = sinθ so that (1) becomes; D 1 = (d/dx)(sin -1 (3sinθ-4sin 3 θ)) ie, D 1 = (d/dx)(sin -1 (sin 3θ) ie, D 1 = (d/dx)(3θ) ie, D 1 = (d/dx) (3 sin -1 x) similarly, putting x = cos φ in (2), we can show that D 2 = (d/dx) (3 cos -1 x) Therefore, f'(x) = D 1 +D 2 = (d/dx) (3 sin -1 x) + (d/dx) (3 cos -1 x) = (d/dx)(3(sin -1 x+ cos -1 x) = (d/dx)(3π÷2) =0 But, We can prove that the aboce given function f(x) is not a constant function either by substituting 3x-4x 3 = y or using a scientific calculator!!! How this is possible? Can the derivative of a non-constant function be zero???

Consider the function f(x) = sin-1(3x-4x3) + cos-1(4x3-3x)


Let the dreivative of the function be, f'(x) = D1+D2


where D1 = (d/dx)(sin-1(3x-4x3))   -------------(1)


and D2 = (d/dx)(cos-1(4x3-3x))    ------------(2)


Let x = sinθ so that (1) becomes;


D1 = (d/dx)(sin-1(3sinθ-4sin3θ))


ie, D1 = (d/dx)(sin-1(sin 3θ)


ie, D1 = (d/dx)(3θ)


ie, D1 = (d/dx) (3 sin-1 x)


similarly, putting x = cos φ in (2), we can show that


D2 = (d/dx) (3 cos-1 x)


 


Therefore, f'(x) = D1+D2


= (d/dx) (3 sin-1 x) + (d/dx) (3 cos-1 x)


= (d/dx)(3(sin-1 x+ cos-1 x)


= (d/dx)(3π÷2)


=0


But, We can prove that the aboce given function f(x) is not a constant function either by substituting 3x-4x3 = y or using a scientific calculator!!!


 


How this is possible? Can the derivative of a non-constant function be zero???

Grade:12

2 Answers

krishna sandilya
8 Points
12 years ago
dude, there is no rule that the derivative of a function is not zero it is always zero for a constant function one thing is that you have definitely gone wrong when you have substituted 3theta for arcsin[sin[theta]] by the way may i know the values of your scientific calculator that has shown dif values
Badiuddin askIITians.ismu Expert
148 Points
12 years ago

Dear Davis Devasia

sin-1 x+ cos-1 x =∏/2

you can use this formula if x is same in both sin-1x and cos-1x


but here you can use it because x in sin-1x is sinΘ  

and x in cos-1x is cosΦ   ,both are different .you can use this formula


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