To find dz/dy, we need to first find dz/dx and then use the chain rule.
Given Z = tan(xy^2), let's find dz/dx:
dz/dx = d(tan(xy^2))/dxUsing the chain rule, dz/dx = sec^2(xy^2) * (y^2)(dx/dx)Simplifying, dz/dx = y^2sec^2(xy^2)
Now, let's find dz/dy:
dz/dy = dz/dx dx/dydz/dy = y^2sec^2(xy^2) xdz/dy = x*y^2sec^2(xy^2)
Now, let's find d^2z/dx^2:
d^2z/dx^2 = d/dx(dy/dx) d/dy(y^2sec^2(xy^2))d^2z/dx^2 = y^2 d/dx(sec^2(xy^2))d^2z/dx^2 = y^2 2sec(xy^2)tan(xy^2) d(xy^2/dx)d^2z/dx^2 = 2y^2sec(xy^2)tan(xy^2) (y^2)
Next, let's find d^2z/d^2y:
d^2z/d^2y = d/dy(dy/dx) d/dy(y^2sec^2(xy^2))d^2z/d^2y = d/dy(dy/dx) 2ysec^2(xy^2)d^2z/d^2y = d/dy(y^2sec^2(xy^2))d^2z/d^2y = 2ysec^2(xy^2)
Therefore, the final results are:
To find dz/dy, we need to first find dz/dx and then use the chain rule.
Given Z = tan(xy^2), let's find dz/dx:
dz/dx = d(tan(xy^2))/dx
Using the chain rule, dz/dx = sec^2(xy^2) * (y^2)(dx/dx)
Simplifying, dz/dx = y^2sec^2(xy^2)
Now, let's find dz/dy:
dz/dy = dz/dx dx/dy
dz/dy = y^2sec^2(xy^2) x
dz/dy = x*y^2sec^2(xy^2)
Now, let's find d^2z/dx^2:
d^2z/dx^2 = d/dx(dy/dx) d/dy(y^2sec^2(xy^2))
d^2z/dx^2 = y^2 d/dx(sec^2(xy^2))
d^2z/dx^2 = y^2 2sec(xy^2)tan(xy^2) d(xy^2/dx)
d^2z/dx^2 = 2y^2sec(xy^2)tan(xy^2) (y^2)
Next, let's find d^2z/d^2y:
d^2z/d^2y = d/dy(dy/dx) d/dy(y^2sec^2(xy^2))
d^2z/d^2y = d/dy(dy/dx) 2ysec^2(xy^2)
d^2z/d^2y = d/dy(y^2sec^2(xy^2))
d^2z/d^2y = 2ysec^2(xy^2)
Therefore, the final results are:
dz/dy = x*y^2sec^2(xy^2)d^2z/dx^2 = 2y^2sec(xy^2)tan(xy^2) * y^2d^2z/d^2y = 2ysec^2(xy^2)