Use the chain rule to find
dw/dt.   w = xe^(y⁄z),   x = t^7,   y = 4 − t,   z = 3 + 4t

To find \(\frac{dw}{dt}\), we'll use the chain rule along with the expressions for \(x\), \(y\), and \(z\). The chain rule states that if \(w = w(x, y, z)\) and \(x = x(t)\), \(y = y(t)\), and \(z = z(t)\), then:

\[
\frac{dw}{dt} = \frac{\partial w}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial w}{\partial y} \cdot \frac{dy}{dt} + \frac{\partial w}{\partial z} \cdot \frac{dz}{dt}
\]

### Step 1: Differentiate \(w = x e^{y/z}\) with respect to each variable

#### Partial derivative of \(w\) with respect to \(x\):

\[
\frac{\partial w}{\partial x} = e^{y/z}
\]

#### Partial derivative of \(w\) with respect to \(y\):

\[
\frac{\partial w}{\partial y} = x \cdot \frac{1}{z} e^{y/z}
\]

#### Partial derivative of \(w\) with respect to \(z\):

\[
\frac{\partial w}{\partial z} = x \cdot e^{y/z} \cdot \left(-\frac{y}{z^2}\right)
\]

### Step 2: Find derivatives of \(x\), \(y\), and \(z\) with respect to \(t\)

- \(x = t^7\), so \(\frac{dx}{dt} = 7t^6\)
- \(y = 4 - t\), so \(\frac{dy}{dt} = -1\)
- \(z = 3 + 4t\), so \(\frac{dz}{dt} = 4\)

### Step 3: Apply the chain rule

Now, apply the chain rule:

\[
\frac{dw}{dt} = \frac{\partial w}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial w}{\partial y} \cdot \frac{dy}{dt} + \frac{\partial w}{\partial z} \cdot \frac{dz}{dt}
\]

Substitute the partial derivatives and the derivatives of \(x\), \(y\), and \(z\) with respect to \(t\):

\[
\frac{dw}{dt} = e^{y/z} \cdot 7t^6 + x \cdot \frac{1}{z} e^{y/z} \cdot (-1) + x \cdot e^{y/z} \cdot \left(-\frac{y}{z^2}\right) \cdot 4
\]

### Step 4: Simplify the expression

Factor out \(e^{y/z}\):

\[
\frac{dw}{dt} = e^{y/z} \left( 7t^6 - \frac{x}{z} - \frac{4xy}{z^2} \right)
\]

Now, substitute \(x = t^7\), \(y = 4 - t\), and \(z = 3 + 4t\) into the equation:

\[
\frac{dw}{dt} = e^{(4 - t)/(3 + 4t)} \left( 7t^6 - \frac{t^7}{3 + 4t} - \frac{4t^7(4 - t)}{(3 + 4t)^2} \right)
\]

This is the simplified expression for \(\frac{dw}{dt}\).

Would you like to go into further details on simplifying this or have any questions?

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**Related questions:**

1. How can the chain rule be used in implicit differentiation?
2. Can you explain the product and quotient rules in calculus?
3. What is the significance of the exponential function \(e^x\) in calculus?
4. How do you solve differential equations involving exponential functions?
5. How can the chain rule be extended to multivariable functions?

**Tip:** When working with the chain rule, always carefully track each variable's dependency to avoid confusion when taking partial derivatives.

Use the chain rule to find dw/dt. w = xe^(y/z), x = t^7, y = 4 − t, z = 3 + 4t

This solution applies the chain rule to find dw/dt for the equation w = xe^(y/z), where x = t^7, y = 4 − t, and z = 3 + 4t. Learn how to differentiate multivariable functions using partial derivatives and the chain rule. Suitable for college-level calculus students.

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