Question #bbb77

2 Answers
Jan 15, 2017

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

Jan 15, 2017

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