find (d^2y)/(dx^2) if x^2-2y^2=3 (d^2y)/(dx^2)=

pramodpandey | Student

Given   `x^2-2y^2=3`     (i)

differentiate with respect to x (implicitly).




`(dy)/(dx)=(1/2)xy^(-1)`       (ii)

differentiate (ii) w.r.t. x ,


Substitute `(dy)/(dx)`from (ii) ,we have


`` `=(1/2)(y^(-1)-(x^2y^(-3))/2)`


`=-3/(4y^3)`       ,   using(i)



oldnick | Student

first we have to  set  funtion in form  y = f(x)

`y=` `sqrt(2)/2` `sqrt(x^2 -3)`

now we have composed function derivative:

`d/dx`  `u[v(x)]= (du)/(dv)v'(x) `

` v(x) = x^2-3 and u(x)= sqrt[v(x)]`

`dy/dx=sqrt(2)/2 [2x ]/[2sqrt(x^2 - 3)]`


`dy/dx=sqrt(2)/2 (x)/sqrt(x^2 -3)`


`(d^2y)/(dx^2) = sqrt(2)/2 [sqrt(x^2 -3)-(x^2)/sqrt(x^2-3)]/[x^2-3)`

`(d^2y)/(dx^2)= sqrt(2)/2 [-3]/[sqrt(x^2 -3) (x^2 -3)]`

`(d^2y)/(dx^2)=-3sqrt(2)/2 1/[(x^2-3)^(3/2)]`


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