91Ó°ÊÓ

Partial derivatives are a crucial concept when dealing with functions of multiple variables. Imagine a function that depends on two or more variables, like our function \( z = e^{1-xy} \) that depends on both \( x \) and \( y \). Instead of finding the effect on \( z \) due to a change in all variables at once, partial derivatives allow us to see how changing one variable, while keeping others constant, affects the function.

In practice, this is like asking: "What happens to \( z \) if only \( x \) changes while \( y \) stays the same?" or vice versa. For our function, we calculated the partial derivative with respect to \( x \), \( \frac{\partial z}{\partial x} \), resulting in \( -y\cdot e^{1-xy} \). This shows that for a small change in \( x \), \( z \) will change by this expression, assuming \( y \) is constant.

Similarly, the partial derivative with respect to \( y \), \( \frac{\partial z}{\partial y} \), gives us \( -x\cdot e^{1-xy} \). With partial derivatives, we can later use these results in the chain rule to find how \( z \) changes over time with respect to another parameter.

The derivative with respect to a parameter is necessary when we are interested in understanding how a dependent variable changes over time or some other changing parameter.

In our example, we needed to find \( \frac{dz}{dt} \), where both \( x \) and \( y \) are functions of \( t \). This requires using the chain rule for differentiation, which helps in dealing with composite functions. The chain rule states:

• \( \frac{dz}{dt} = \frac{\partial z}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial z}{\partial y} \cdot \frac{dy}{dt} \)

This relationship uses the partial derivatives \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \) that we derived earlier, alongside the derivatives \( \frac{dx}{dt} = \frac{1}{3}t^{-2/3} \) and \( \frac{dy}{dt} = 3t^2 \), which reflect the change in \( x \) and \( y \) with \( t \).

By plugging these into the chain rule format, one can compute \( \frac{dz}{dt} \) effectively, thus understanding how \( z \) evolves as \( t \) varies.

Exponential functions, such as \( e^{1-xy} \) in our equation for \( z \), are powerful mathematical tools that describe situations where growth or decay is proportional to the current amount.

The base of natural exponential functions is \( e \), approximately equal to 2.718. It is unique because its derivative is the same as the function itself, which provides simplicity when deriving expressions that involve \( e \).

Within our problem, the exponential function evaluates the expression \( 1-xy \), which scales the impact of the exponential behavior based on the values of \( x \) and \( y \). In essence, the exponent affects how the function grows or decays based on the combined interaction of \( x \) and \( y \).

• For instance, when differentiating, the exponential itself remains, while we take into account the derivative of the exponent (like \(-xy\) in our partial derivatives).

The beauty of these functions is revealed through applications ranging from biology to economics, offering insights into trends and changes in complex systems.