We are given the equation of an astroid:

$x^{2/3} + y^{2/3} = 4$

We need to find the equation of the tangent line to this curve at the point $\left(-3\sqrt{3}, 1\right)$ using implicit differentiation.

Step 1: Differentiate the equation implicitly

Differentiate both sides of the given equation with respect to $x$, keeping in mind that $y$ is a function of $x$.

$\frac{d}{dx}\left(x^{2/3} + y^{2/3}\right) = \frac{d}{dx}(4)$

On the left-hand side, apply the chain rule:

$\frac{d}{dx}(x^{2/3}) = \frac{2}{3}x^{-1/3}$ $\frac{d}{dx}(y^{2/3}) = \frac{2}{3}y^{-1/3} \cdot \frac{dy}{dx}$

Thus, the differentiated equation becomes:

$\frac{2}{3}x^{-1/3} + \frac{2}{3}y^{-1/3} \cdot \frac{dy}{dx} = 0$

Step 2: Solve for $\frac{dy}{dx}$

Now solve this equation for $\frac{dy}{dx}$:

$\frac{2}{3}y^{-1/3} \cdot \frac{dy}{dx} = - \frac{2}{3}x^{-1/3}$

Cancel out the $\frac{2}{3}$ on both sides:

$y^{-1/3} \cdot \frac{dy}{dx} = - x^{-1/3}$

Now isolate $\frac{dy}{dx}$:

$\frac{dy}{dx} = - \frac{x^{-1/3}}{y^{-1/3}}$

This simplifies to:

$\frac{dy}{dx} = - \left( \frac{x}{y} \right)^{1/3}$

Step 3: Find the slope of the tangent line at the point $(-3\sqrt{3}, 1)$

Substitute $x = -3\sqrt{3}$ and $y = 1$ into the expression for $\frac{dy}{dx}$:

$\frac{dy}{dx} = - \left( \frac{-3\sqrt{3}}{1} \right)^{1/3} = - \left( -3\sqrt{3} \right)^{1/3}$

We need to simplify $(-3\sqrt{3})^{1/3}$, which is a complex operation. Let's approximate it step-by-step:

$(-3\sqrt{3})^{1/3} \approx -1.9129$

Thus, the slope at the point $(-3\sqrt{3}, 1)$ is approximately:

$\frac{dy}{dx} \approx -(-1.9129) = 1.9129$

Step 4: Find the equation of the tangent line

Using the point-slope form of the equation of a line:

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

Substitute the point $(-3\sqrt{3}, 1)$ and the slope $m \approx 1.9129$:

$y - 1 = 1.9129(x + 3\sqrt{3})$

This is the equation of the tangent line to the curve at the given point.

Would you like me to clarify any details or go deeper into any part of this problem?

Here are 5 related questions for further practice:

1. How does implicit differentiation differ from explicit differentiation?
2. What is the general method to differentiate an equation implicitly?
3. How would the process change if we had a different exponent in the astroid equation?
4. Can you find the normal line equation to this curve at the same point?
5. How would you compute the slope if the point had complex coordinates?

Tip: Always check the differentiability of implicit functions at a point before finding the slope of the tangent line.