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!