91Ó°ÊÓ

Partial derivatives allow us to see how a function changes as one of its variables is changed, while keeping all other variables constant. In the original problem, the function \( z = \ln(2x^2 + y) \) is defined with variables \( x \) and \( y \). Since both \( x \) and \( y \) are also functions of \( t \), partial derivatives help us find how the function \( z \) changes with respect to each of these independent variables.

To compute partial derivatives of \( z \) with respect to \( x \) and \( y \), we treat each one as the variable of differentiation while treating the other as a constant. For instance:

• The partial derivative of \( z \) with respect to \( x \), denoted as \( \frac{\partial z}{\partial x} \), involves differentiating \( \ln(2x^2 + y) \) by \( x \), giving us \( \frac{4x}{2x^2 + y} \).


• Similarly, the partial derivative of \( z \) with respect to \( y \), \( \frac{\partial z}{\partial y} \), is computed as \( \frac{1}{2x^2 + y} \).

These partial derivatives are crucial for applying the chain rule, which calculates total derivatives like \( \frac{dz}{dt} \).

When dealing with differentiation, functions of a variable refer to mathematical expressions where one variable depends on another. In this context, both \( x \) and \( y \) are functions of \( t \).

• For \( x = \sqrt{t} \), as \( t \) changes, \( x \) changes accordingly, reflecting its dependency on \( t \).


• Similarly, \( y = t^{2/3} \) implies that any variation in \( t \) results in a corresponding variation in \( y \).

The task is to trace how changes in \( t \) affect \( x \), \( y \), and ultimately \( z \). Recognizing these dependencies allows for the proper application of the chain rule to calculate \( \frac{dz}{dt} \).

Differentiation with respect to a variable involves identifying how a function changes with that variable. Given that \( z \) depends on \( x \) and \( y \), both of which are influenced by \( t \), differentiation here requires the chain rule. This method is effective in multi-variable functions, linking rate of change in nested and composed functions.

The chain rule formula for \( \frac{dz}{dt} \) is:
\[ \frac{dz}{dt} = \frac{\partial z}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial z}{\partial y} \cdot \frac{dy}{dt} \]

• The derivative \( \frac{dx}{dt} = \frac{1}{2\sqrt{t}} \), calculated using \( x = \sqrt{t} \), represents the rate of change of \( x \) with respect to time \( t \).


• Similarly, \( \frac{dy}{dt} = \frac{2}{3}t^{-1/3} \) shows how \( y \) changes over time \( t \).

By substituting these derivatives into the chain rule equation, we can find the overall effect on \( z \) as \( t \) changes, completing the differentiation process.