If `y=y(x)` and it follows the relation `4x e^(x y)=y+5sin^2x ,` then `y^(prime)(0)` is equal to______

Explanation

We have `4xe^(xy)=y+5 sin^(2)x`
Putting, x = 0 in above equation we get y=0
`therefore" (0,0) lies on the curve"`
Now on differentaiting equation (1) w.r.t.x, we get
`4ee^(xy)+4xe^(xy)(x(dy)/(dx)+y)=(dy)/(dx)+10 sin x cos x`
Putting x= 0 and y=0, we get
`4+0=((dy)/(dx))_("(0,0)")+0`
`therefore" "((dy)/(dx))_("(0,0)")=4`