Rates of change?

Question

If z=f(x,y) and z=3x^2y^5 , find the rate of change of z when x is 3 units and y is 2 units when x is decreasing at 5 units/s and y is increasing at 2.5 units/s


Help please walk through this slowly thanks

Answer 1

dz = f_xdx +f_ydy
f_x= 6xy^5
f_y=15x^2y^4
x=3, y=2
dx/dt=-5 and dy/dt= 2.5

dz/dt= 6*3*2^5(-5) +15(3)^2*(2)^4 (2.5)=2520units/s

Answer 2

z = 3 x^2 y^5

z' = 3 (x^2)' y^5 + 3 x^2 (y^5)'
= 3*2xx' y^5 +3*4y^4y' x^2
= 3 x y^4 ( 2yx' + 4xy' )

= 3*3*2^4( 2*2*(-5) +4*3*2.5 ) ... at the time in question
= 144 ( -20 + 30 )
= 1440