Let y^3+x^2y^5=e^4x.
Use implicit differentiation to find a formula for  dy/dx

We are given the equation:

\[
y^3 + x^2 y^5 = e^{4x}
\]

To find \(\frac{dy}{dx}\), we will differentiate both sides with respect to \(x\), applying implicit differentiation. Since \(y\) is a function of \(x\), we'll need to use the chain rule when differentiating any term involving \(y\).

### Step-by-Step Differentiation:

1. Differentiate the first term \(y^3\) with respect to \(x\):

\[
\frac{d}{dx}(y^3) = 3y^2 \frac{dy}{dx}
\]

2. Differentiate the second term \(x^2 y^5\) using the product rule. The product rule states that:

\[
\frac{d}{dx}(u v) = u'v + uv'
\]

Let \(u = x^2\) and \(v = y^5\). Then:

\[
\frac{d}{dx}(x^2 y^5) = \frac{d}{dx}(x^2) \cdot y^5 + x^2 \cdot \frac{d}{dx}(y^5)
\]

Now, differentiate each part:

\[
\frac{d}{dx}(x^2) = 2x
\]

\[
\frac{d}{dx}(y^5) = 5y^4 \frac{dy}{dx}
\]

Thus, we get:

\[
\frac{d}{dx}(x^2 y^5) = 2x y^5 + x^2 \cdot 5y^4 \frac{dy}{dx}
\]

3. Differentiate the right-hand side \(e^{4x}\) with respect to \(x\):

\[
\frac{d}{dx}(e^{4x}) = 4e^{4x}
\]

### Now, putting it all together:

The differentiated equation becomes:

\[
3y^2 \frac{dy}{dx} + \left(2x y^5 + 5x^2 y^4 \frac{dy}{dx}\right) = 4e^{4x}
\]

### Isolating \(\frac{dy}{dx}\):

Move the terms without \(\frac{dy}{dx}\) to the right side:

\[
3y^2 \frac{dy}{dx} + 5x^2 y^4 \frac{dy}{dx} = 4e^{4x} - 2x y^5
\]

Factor out \(\frac{dy}{dx}\) on the left side:

\[
\frac{dy}{dx}(3y^2 + 5x^2 y^4) = 4e^{4x} - 2x y^5
\]

Finally, solve for \(\frac{dy}{dx}\):

\[
\frac{dy}{dx} = \frac{4e^{4x} - 2x y^5}{3y^2 + 5x^2 y^4}
\]

This is the formula for \(\frac{dy}{dx}\).

Would you like more details or have any questions?

### Related questions:
1. How is implicit differentiation used for higher-order derivatives?
2. What is the geometric interpretation of \(\frac{dy}{dx}\) in this equation?
3. Can this formula for \(\frac{dy}{dx}\) be simplified further?
4. How would you apply implicit differentiation to trigonometric functions?
5. What role does the chain rule play in implicit differentiation?

**Tip:** Always apply the product rule carefully when differentiating terms involving two functions of \(x\).

Let y^3 + x^2 y^5 = e^{4x}. Use implicit differentiation to find a formula for dy/dx.

Learn how to apply implicit differentiation to solve the equation y^3 + x^2 y^5 = e^{4x} and find the formula for dy/dx. This solution covers the chain rule, product rule, and exponential differentiation. Suitable for students in advanced calculus.

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