Using Chain rule, find ∂f/∂t | (t,w)=(1,1). f(x,y)=e^(xy^3) , x=t^2+w^2, y=tw^2.

To find \(\frac{\partial f}{\partial t} \bigg|_{(t,w)=(1,1)}\) using the Chain Rule, let's follow these steps:

1. **Express \(f\) in terms of \(t\) and \(w\):**
   \[
   f(x, y) = e^{xy^3}
   \]
   where
   \[
   x = t^2 + w^2 \quad \text{and} \quad y = tw^2.
   \]

2. **Compute the partial derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\):**
   \[
   \frac{\partial f}{\partial x} = \frac{d}{dx}\left(e^{xy^3}\right) = y^3 e^{xy^3},
   \]
   \[
   \frac{\partial f}{\partial y} = \frac{d}{dy}\left(e^{xy^3}\right) = 3xy^2 e^{xy^3}.
   \]

3. **Compute the partial derivatives \(\frac{\partial x}{\partial t}\) and \(\frac{\partial y}{\partial t}\):**
   \[
   \frac{\partial x}{\partial t} = \frac{d}{dt}(t^2 + w^2) = 2t,
   \]
   \[
   \frac{\partial y}{\partial t} = \frac{d}{dt}(tw^2) = w^2.
   \]

4. **Apply the Chain Rule:**
   \[
   \frac{\partial f}{\partial t} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial t}.
   \]

5. **Substitute the values of the partial derivatives and the expressions for \(f\), \(x\), and \(y\):**
   \[
   \frac{\partial f}{\partial t} = (y^3 e^{xy^3}) (2t) + (3xy^2 e^{xy^3}) (w^2).
   \]

6. **Evaluate at \(t = 1\) and \(w = 1\):**
   \[
   x = 1^2 + 1^2 = 2,
   \]
   \[
   y = 1 \cdot 1^2 = 1.
   \]
   Therefore,
   \[
   \frac{\partial f}{\partial x} \bigg|_{(t,w)=(1,1)} = 1^3 e^{2 \cdot 1^3} = e^2,
   \]
   \[
   \frac{\partial f}{\partial y} \bigg|_{(t,w)=(1,1)} = 3 \cdot 2 \cdot 1^2 e^{2 \cdot 1^3} = 6e^2.
   \]
   Also,
   \[
   \frac{\partial x}{\partial t} \bigg|_{(t,w)=(1,1)} = 2 \cdot 1 = 2,
   \]
   \[
   \frac{\partial y}{\partial t} \bigg|_{(t,w)=(1,1)} = 1^2 = 1.
   \]

7. **Combine all these values:**
   \[
   \frac{\partial f}{\partial t} \bigg|_{(t,w)=(1,1)} = (e^2)(2) + (6e^2)(1) = 2e^2 + 6e^2 = 8e^2.
   \]

So, \(\frac{\partial f}{\partial t} \bigg|_{(t,w)=(1,1)} = 8e^2\).

Would you like more details or have any questions?

Here are 5 more questions you can ask:
1. How do you find the second derivative using the chain rule?
2. Can you explain the product rule with an example?
3. What is the integral of \(e^{xy}\) with respect to \(x\)?
4. How do you solve differential equations using the chain rule?
5. What is the gradient of a multivariable function?

**Tip:** Always break down complex problems into smaller parts to simplify the solution process.

Learn how to use the Chain Rule to find partial derivatives in multivariable calculus with a step-by-step example. This problem involves the function f(x,y)=e^(xy^3) with variables x=t^2+w^2 and y=tw^2, evaluated at (t,w)=(1,1). Explore the calculation process and understand the application of the Chain Rule in calculus.

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