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We are tasked with using implicit differentiation to find \(\frac{dy}{dx}\) for the given equation:

\[
e^{4x\sqrt{y}} = y^5.
\]

### Step-by-Step Solution:

1. **Differentiate both sides with respect to \(x\):**

   We'll apply the chain rule, and keep in mind that \(y\) is a function of \(x\), so when differentiating terms involving \(y\), we must multiply by \(\frac{dy}{dx}\).

   For the left-hand side:
   \[
   \frac{d}{dx} \left(e^{4x\sqrt{y}}\right) = e^{4x\sqrt{y}} \cdot \frac{d}{dx}(4x\sqrt{y}).
   \]
   Now, differentiate \(4x\sqrt{y}\):
   \[
   \frac{d}{dx}(4x\sqrt{y}) = 4\sqrt{y} + 4x \cdot \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} = 4\sqrt{y} + \frac{2x}{\sqrt{y}} \cdot \frac{dy}{dx}.
   \]
   So the derivative of the left-hand side becomes:
   \[
   e^{4x\sqrt{y}} \left(4\sqrt{y} + \frac{2x}{\sqrt{y}} \cdot \frac{dy}{dx}\right).
   \]

2. **Differentiate the right-hand side:**
   \[
   \frac{d}{dx} \left(y^5\right) = 5y^4 \cdot \frac{dy}{dx}.
   \]

3. **Set up the equation:**
   Now we have:
   \[
   e^{4x\sqrt{y}} \left(4\sqrt{y} + \frac{2x}{\sqrt{y}} \cdot \frac{dy}{dx}\right) = 5y^4 \cdot \frac{dy}{dx}.
   \]

4. **Solve for \(\frac{dy}{dx}\):**

   First, expand the left-hand side:
   \[
   e^{4x\sqrt{y}} \cdot 4\sqrt{y} + e^{4x\sqrt{y}} \cdot \frac{2x}{\sqrt{y}} \cdot \frac{dy}{dx} = 5y^4 \cdot \frac{dy}{dx}.
   \]
   Now, isolate the terms involving \(\frac{dy}{dx}\):
   \[
   e^{4x\sqrt{y}} \cdot \frac{2x}{\sqrt{y}} \cdot \frac{dy}{dx} - 5y^4 \cdot \frac{dy}{dx} = -e^{4x\sqrt{y}} \cdot 4\sqrt{y}.
   \]
   Factor out \(\frac{dy}{dx}\) from the left-hand side:
   \[
   \frac{dy}{dx} \left(e^{4x\sqrt{y}} \cdot \frac{2x}{\sqrt{y}} - 5y^4\right) = -e^{4x\sqrt{y}} \cdot 4\sqrt{y}.
   \]
   Finally, solve for \(\frac{dy}{dx}\):
   \[
   \frac{dy}{dx} = \frac{-e^{4x\sqrt{y}} \cdot 4\sqrt{y}}{e^{4x\sqrt{y}} \cdot \frac{2x}{\sqrt{y}} - 5y^4}.
   \]
   You can simplify the numerator further if necessary, but this is the implicit derivative expression.

### Would you like a further breakdown of any step or have any questions?

#### Here are 5 additional related questions to consider:
1. How do we apply implicit differentiation when multiple variables are involved?
2. What are the conditions under which we can apply the chain rule?
3. Can implicit differentiation be used to find second-order derivatives as well?
4. How would this problem change if the exponent in \(y^5\) were different?
5. Can we solve for \(y(x)\) explicitly, or is implicit differentiation the only method?

**Tip:** Always remember to use the chain rule when differentiating composite functions, especially when implicit differentiation is involved.

Use implicit differentiation to find dy/dx when x and y satisfy e^(4x * sqrt(y)) = y^5.

Learn how to solve the implicit differentiation problem e^(4x * sqrt(y)) = y^5 using chain rule and the power rule for derivatives. This problem involves applying advanced calculus concepts suitable for high school or early college students.

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