Assume that z=f(x, y), x=g(t), y=h(t), f'(2,-1)=3 and fy(2,-1)=-2 . If g(

step 1 In this question, we're given z is a function of x and y, where x and y are both functions of t.

Assume that z=f(x, y), x=g(t), y=h(t), ...

$$ \begin{array}{l}{\text { Assume that } z=f(x, y), \quad x=g(t), \quad y=h(t), \quad f_{\prime}(2,-1)=3} \\ {\text { and } f_{y}(2,-1)=-2 . \text { If } g(0)=2, h(0)=-1, g^{\prime}(0)=5, \text { and }} \\ {h^{\prime}(0)=-4, \text { find }\left.\frac{d z}{d t}\right|_{t=0}}\end{array} $$

$$
\begin{array}{l}{\text { Assume that } z=f(x, y), \quad x=g(t), \quad y=h(t), \quad f_{\prime}(2,-1)=3} \\ {\text { and } f_{y}(2,-1)=-2 . \text { If } g(0)=2, h(0)=-1, g^{\prime}(0)=5, \text { and }} \\ {h^{\prime}(0)=-4, \text { find }\left.\frac{d z}{d t}\right|_{t=0}}\end{array}
$$

Key Concepts

Chain Rule for Multivariable Functions

This concept involves taking the derivative of a function that depends on multiple variables, each of which is itself a function of another variable. When a function z = f(x, y) is given with x and y as functions of t, the chain rule allows the derivative dz/dt to be calculated by summing the products of the partial derivatives of f with respect to x and y, and the respective derivatives of x and y with respect to t. This is fundamental in linking the rates of change through composite functions.

Partial Derivatives

Partial derivatives represent the rate at which a function changes as one of its variables changes while keeping the other variables constant. In the context of multivariable functions, they provide a way to understand and compute the sensitivity of the function in each independent direction. They are essential in applying the chain rule to calculate derivatives of composite functions where the output depends on several intermediate variables.

Composite Functions

Composite functions occur when one function is applied to the result of another function. In problems where z = f(g(t), h(t)), understanding how to differentiate composite functions is crucial. The process involves differentiating the outer function with respect to its variables and then multiplying by the derivative of the inner functions with respect to t, as highlighted by the chain rule for multivariable functions.

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