Question

Given, \(z(x, y) = e^{x - 2y}\), where \(x(t) = e^t\) and \(y(t) = e^{-t}\). All the variables are real. The total differential \(\frac{dz}{dt}\) is

• A. $-z(x + 2y)$
• B. $-z(x - 2y)$
• C. $z(x + 2y)$
• D. $z(x - 2y)$

Hint:

To find the total differential, use the chain rule by multiplying the partial derivatives with respect to each variable and the corresponding time derivatives.

Answer 1

The Correct Option is C

We are given the function $z(x, y) = e^{x - 2y}$. To find the total differential $\frac{dz}{dt}$, we use the chain rule. First, we calculate the partial derivatives of $z$ with respect to $x$ and $y$: \[ \frac{\partial z}{\partial x} = e^{x - 2y}, \frac{\partial z}{\partial y} = -2e^{x - 2y} \] Now, apply the chain rule: \[ \frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt} \] Substitute the values of $\frac{dx}{dt} = e^t$ and $\frac{dy}{dt} = -e^{-t}$ into the equation: \[ \frac{dz}{dt} = e^{x - 2y} \cdot e^t + (-2e^{x - 2y}) \cdot (-e^{-t}) = e^{x - 2y}(e^t + 2e^{-t}) \] Thus, the total differential is: \[ \frac{dz}{dt} = z(x + 2y) \] Therefore, the correct answer is (C).