It's #vec(grad)p(x,y) = langle-(4x)/(sqrt(24-4x^2-y^2)) , -(y)/(sqrt(24-4x^2-y^2)) rangle#
The gradient #vec(grad)p(x,y)# of the function #p(x,y)# is defined as the vector whose #n#-th component is the partial derivative of #p# with respect to the #n#-th variable, or:
#vec(grad)p(x,y) = langle (del p)/(del x), (del p)/(del y) rangle#
Computing the partial derivatives using the chain rule:
#(del p)/(del x) = (del)/(del x)[sqrt(24-4x^2-y^2)] #
#= 1/(2sqrt(24-4x^2-y^2)) (del)/(del x)[24-4x^2-y^2] #
#= -(4x)/(sqrt(24-4x^2-y^2))#
#(del p)/(del y) = (del)/(del y)[sqrt(24-4x^2-y^2)]#
#= 1/(2sqrt(24-4x^2-y^2)) (del)/(del y)[24-4x^2-y^2] #
#= -(y)/(sqrt(24-4x^2-y^2))#
And so,
#vec(grad)p(x,y) = langle (del p)/(del x), (del p)/(del y) rangle#
#= langle -(4x)/(sqrt(24-4x^2-y^2)) , -(y)/(sqrt(24-4x^2-y^2)) rangle#
