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The given equation is:

\[
y = \sqrt[3]{1 + 7x}
\]

To find \(\frac{dy}{dx}\), we need to differentiate the equation with respect to \(x\).

### Step-by-Step Differentiation:

1. **Rewrite the function in a more convenient form**:
   \[
   y = (1 + 7x)^{\frac{1}{3}}
   \]

2. **Use the chain rule**: 
   \[
   \frac{dy}{dx} = \frac{d}{dx} \left( (1 + 7x)^{\frac{1}{3}} \right)
   \]
   First, differentiate the outer function \((u)^{\frac{1}{3}}\) where \(u = 1 + 7x\):
   \[
   \frac{d}{du} \left( u^{\frac{1}{3}} \right) = \frac{1}{3} u^{-\frac{2}{3}}
   \]
   Now multiply by the derivative of the inner function \(u = 1 + 7x\):
   \[
   \frac{d}{dx}(1 + 7x) = 7
   \]

3. **Apply the chain rule**:
   \[
   \frac{dy}{dx} = \frac{1}{3} (1 + 7x)^{-\frac{2}{3}} \times 7
   \]

4. **Simplify**:
   \[
   \frac{dy}{dx} = \frac{7}{3} (1 + 7x)^{-\frac{2}{3}}
   \]

Thus, the derivative is:

\[
\frac{dy}{dx} = \frac{7}{3} (1 + 7x)^{-\frac{2}{3}}
\]

Would you like more details or further clarification?

Here are 5 related questions for further exploration:
1. What is the second derivative of \(y = \sqrt[3]{1 + 7x}\)?
2. How does the derivative \(\frac{dy}{dx}\) behave as \(x\) approaches infinity?
3. How do we interpret the derivative of a cube root function graphically?
4. Can you apply this differentiation technique to other cube root functions like \(y = \sqrt[3]{3x + 5}\)?
5. What is the physical meaning of this derivative in real-world problems?

**Tip**: Whenever you're working with roots or powers, converting them into exponent form often makes differentiation much easier, especially when applying the chain rule!

Find the derivative dy/dx for the function y = ∛(1 + 7x).

Learn how to find the derivative of the function y = ∛(1 + 7x) using the chain rule. This step-by-step solution includes rewriting the function in exponent form and applying the chain rule to differentiate. Suitable for high school or early college students.

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