Question

Question #bbb77

Answer 1

`y=sin(pi/6 e^(xy))`

Differentiating w.r to x we get

`(dy)/(dx)=cos(pi/6e^(xy))xxpi/6e^(xy)xx(y+x(dy)/(dx))`

`(dy)/(dx)=cos(pi/6e^(xy))xxpi/6e^(xy)xx(sin(pi/6e^(xy))+x(dy)/(dx))`

Putting `x=0`

`((dy)/(dx))_(x=0)=cos(pi/6e^(0*y))xxpi/6e^(0*y)xx(sin(pi/6)e^(0*y))+0*(dy)/(dx))`

`=cos(pi/6)xxpi/6xxsin(pi/6)`

`=sqrt3/2xxpi/6xx1/2`

`=(sqrt3pi)/24`

Answer 2

`y'(0) = (pisqrt(3))/24`

Explanation:

We have the equation:

`y=sin(pi/6e^(xy))`

differentiate with respect to `x`:

`y' = (y+xy')pi/6e^(xy)cos(pi/6e^(xy))`

Solving for `y'`:

`y'= frac ( pi/6 y e^(xy)cos(pi/6e^(xy))) (1-pi/6xe^(xy)cos(pi/6e^(xy)))`

Now, for `x=0` we have:

`y(0) = sin((pi/6) e^(0*y) ) = sin(pi/6) =1/2`

and then:

`y'(0) = frac ( pi/6 1/2 cos(pi/6)) (1)=(pisqrt(3))/24`