) If 𝑓 (𝑥, 𝑦) = 𝑒

𝑥𝑦, where 𝑥 = 4/(2t+1)

and 𝑦 = 3𝑡 + 5. Find the total derivative dd𝑡𝑓

at 𝑡 = 0.

To find the total derivative \( \frac{d}{dt} f \) at \( t = 0 \), where \( f(x, y) = e^{xy} \), \( x = \frac{4}{2t+1} \), and \( y = 3t + 5 \), we can again use the chain rule for multivariable functions. 

The total derivative of \( f \) with respect to \( t \) is given by:
\[
\frac{d}{dt} f(x, y) = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}
\]

### Step 1: Calculate \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \)
\[
\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left( e^{xy} \right) = y e^{xy}
\]
\[
\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left( e^{xy} \right) = x e^{xy}
\]

### Step 2: Calculate \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \)
Given:
\[
x = \frac{4}{2t + 1}
\quad \text{and} \quad
y = 3t + 5
\]

To find \( \frac{dx}{dt} \), differentiate \( x \) with respect to \( t \):
\[
\frac{dx}{dt} = \frac{d}{dt} \left( \frac{4}{2t + 1} \right)
\]
Using the chain rule:
\[
\frac{dx}{dt} = -\frac{4 \cdot 2}{(2t + 1)^2} = -\frac{8}{(2t + 1)^2}
\]

To find \( \frac{dy}{dt} \):
\[
\frac{dy}{dt} = \frac{d}{dt} \left( 3t + 5 \right) = 3
\]

### Step 3: Evaluate the functions and their derivatives at \( t = 0 \)
At \( t = 0 \):
\[
x = \frac{4}{2(0) + 1} = 4
\]
\[
y = 3(0) + 5 = 5
\]
\[
\frac{dx}{dt} = -\frac{8}{(2(0) + 1)^2} = -8
\]
\[
\frac{dy}{dt} = 3
\]

Now, compute the partial derivatives of \( f \):
\[
\frac{\partial f}{\partial x} = y e^{xy} = 5e^{4 \cdot 5} = 5e^{20}
\]
\[
\frac{\partial f}{\partial y} = x e^{xy} = 4e^{4 \cdot 5} = 4e^{20}
\]

### Step 4: Compute the total derivative
\[
\frac{d}{dt} f = \left( 5e^{20} \right) \cdot (-8) + \left( 4e^{20} \right) \cdot 3 = -40e^{20}
\]

Thus, the total derivative \( \frac{d}{dt} f \) at \( t = 0 \) is \( -40e^{20} \).

Would you like more details or have any questions? Here are some related questions you might be interested in:

1. How does the chain rule apply to functions that include fractions or other complex expressions?
2. What are the implications of a negative total derivative in this context?
3. Can you explain how the partial derivatives \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) contribute to the total derivative?
4. How does the form of \( x(t) \) affect the behavior of \( f(x(t), y(t)) \) as \( t \) changes?
5. What techniques can be used to verify the correctness of a total derivative calculation?

**Tip:** When differentiating functions that are expressed as fractions, be mindful of using the chain rule or quotient rule appropriately to avoid errors.

Learn how to calculate the total derivative of the function f(x, y) = e^(xy) with respect to t, where x = 4/(2t+1) and y = 3t + 5. Detailed step-by-step solution using the chain rule for multivariable functions.

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