91Ó°ÊÓ

Partial derivatives are at the heart of differential calculus. They help us understand how a multivariable function changes when one of the variables changes, while the others remain constant.
In our given function, \( z = \ln(x^2 y) \), partial derivatives allow us to see how \( z \) is affected by changes in either \( x \) or \( y \) independently.
To find the partial derivative with respect to \( x \), \( x \) is considered a variable, and \( y \) is treated as a constant. Similarly, when finding the partial derivative with respect to \( y \), \( y \) is variable and \( x \) is constant.
This process essentially "slices" the function along one of its input axes, giving us a derivative that reflects the rate of change in one direction.

• The partial derivative with respect to \( x \) is \( \frac{2}{x} \).
• The partial derivative with respect to \( y \) is \( \frac{1}{y} \).

This shows how sensitive \( z \) is to small changes in \( x \) or \( y \). Understanding these derivatives is critical for calculating the total differential, which gives an approximation of how the entire function changes.

The total differential of a function gives an approximation of how the function's output changes in response to small changes in its inputs. It combines the effects of partial derivatives, painting a broader picture of change.
For a function \( z = f(x, y) \), the total differential \( dz \) can be calculated using:
\[ dz = \frac{\partial z}{\partial x} \cdot dx + \frac{\partial z}{\partial y} \cdot dy \]
This equation tells us that the change in \( z \) depends on both \( dx \) and \( dy \), which are the changes in \( x \) and \( y \) respectively.
In our example, we calculated \( dz \) using values of partial derivatives at point \( P \) and small changes \( dx = 0.02 \) and \( dy = -0.04 \). This gave an approximate change \( dz = -0.03 \).

• This calculation uses linear approximation to estimate change over a small region close to point \( P \).
• The result helps quickly predict the behavior of \( z \) without needing complex computations.

This tool is especially powerful in real-world applications, helping us understand potential outcomes with minimal calculations.

Exact change calculation is the process of determining the true difference between the values of a function at two distinct points. Unlike the total differential, which is an approximation, this involves directly calculating the function's output at each point.
To calculate the exact change \( \Delta z \), you find \( z \) at both the initial point \( P \) and the new point \( Q \). The difference \( \Delta z = z_Q - z_P \) gives the exact amount \( z \) has changed.
For \( z = \ln(x^2 y) \), we calculated:

• At \( P(-2, 4) \), \( z_P = \ln(16) \approx 2.7726 \).
• At \( Q(-1.98, 3.96) \), \( z_Q = \ln(15.6352) \approx 2.7495 \).

The exact change \( \Delta z = 2.7495 - 2.7726 = -0.0231 \).
This exact value shows us precisely how much \( z \) changes as we move from \( P \) to \( Q \). It’s essential for validating the accuracy of our total differential, confirming if our approximation was close to the real outcome.