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#