The problem asks for the equation of the tangent line to the curve $y = \sqrt[3]{5x} + 6$ at $x = 2$.

Steps:

1. Find the derivative: The slope of the tangent line is given by the derivative $y'$ of the function at $x = 2$. The function is:

   $y = \sqrt[3]{5x} + 6 = (5x)^{1/3} + 6$

   To differentiate this, we use the chain rule:

   $y' = \frac{d}{dx} \left( (5x)^{1/3} \right)$

   Applying the chain rule:

   $y' = \frac{1}{3}(5x)^{-2/3} \cdot 5 = \frac{5}{3}(5x)^{-2/3}$

2. Evaluate the derivative at $x = 2$: Now, substitute $x = 2$ into the derivative:

   $y'(2) = \frac{5}{3}(5 \cdot 2)^{-2/3} = \frac{5}{3} \cdot 10^{-2/3}$

   Simplify $10^{-2/3}$:

   $10^{-2/3} = \frac{1}{10^{2/3}}$

   So, the slope of the tangent line at $x = 2$ is:

   $m = \frac{5}{3 \cdot 10^{2/3}}$

3. Find the point on the curve: To find the point where the tangent touches the curve at $x = 2$, substitute $x = 2$ into the original function:

   [ y(2) = \sqrt[3]{5 \cdot 2} + 6 = \sqrt[3]{10} + 6 ]

   Therefore, the point is $(2, \sqrt[3]{10} + 6)$.

4. Equation of the tangent line: Using the point-slope form of the equation of a line:

   $y - y_1 = m(x - x_1)$

   Substituting $m = \frac{5}{3 \cdot 10^{2/3}}$, $x_1 = 2$, and $y_1 = \sqrt[3]{10} + 6$:

   $y - (\sqrt[3]{10} + 6) = \frac{5}{3 \cdot 10^{2/3}}(x - 2)$

This is the equation of the tangent line at $x = 2$.

Would you like more details on any of these steps or explanations?

Here are 5 related questions for further practice:

1. How would the derivative change if the original function were $y = \sqrt[4]{5x} + 6$?
2. What is the general formula for the tangent line at any point $x = a$ on the curve $y = \sqrt[3]{5x} + 6$?
3. What is the geometric interpretation of the derivative at a point on a curve?
4. How do we find the normal line (perpendicular) to a curve at a given point?
5. What happens to the slope of the tangent line as $x \to 0$ for the function $y = \sqrt[3]{5x} + 6$?

Tip: Always check the units of each term in the derivative to avoid simplification errors when applying the chain rule!