Assume that z = f(x, y)^2, x = g(t), y = h(t), f_y(1, 0) = 1, f_x(1, 0) = -1, f(1, 0) = 2. If g(3) = 1, h(3) = 0, g'(3) = -3, and h'(3) = 4, find dz/dt|_{t=3}.

Steps to Solve

1. Identify the multivariable function and its derivatives
   Given that ( z = f(x, y)^2 ), ( x = g(t) ), and ( y = h(t) ), we will need to use the chain rule to differentiate ( z ) with respect to ( t ). We are given the derivatives ( f_x(1, 0), f_y(1, 0), g'(3), ) and ( h'(3) ) along with specific function values.

2. Apply the Chain Rule
   Using the chain rule, we can express ( \frac{dz}{dt} ) as follows:
   $$ \frac{dz}{dt} = \frac{\partial z}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial z}{\partial y} \cdot \frac{dy}{dt} $$
   Substituting in the partial derivatives: $$ \frac{dz}{dt} = f_x(x,y) \cdot g'(t) + f_y(x,y) \cdot h'(t) $$

3. Evaluate at the specific point
   At ( t = 3 ), we find:

• ( g(3) = 1 ) and ( h(3) = 0 )
  thus, substituting ( x = 1 ) and ( y = 0 ): $$ \frac{dz}{dt} \bigg|_{t=3} = f_x(1, 0) \cdot g'(3) + f_y(1, 0) \cdot h'(3) $$
• The derivatives are given as ( f_x(1, 0) = -1 ), ( f_y(1, 0) = 1 ), ( g'(3) = -3 ), and ( h'(3) = 4 ).

4. Substituting values into the equation
   Substituting these values into the equation: $$ \frac{dz}{dt} \bigg|_{t=3} = (-1)(-3) + (1)(4) $$

5. Calculate the final value
   Calculating: $$ \frac{dz}{dt} \bigg|_{t=3} = 3 + 4 = 7 $$