https://file.aiquickdraw.com/mathbot/file/1728705291822qtq2i54m.jpg

Let's break down and solve the problem shown in the image.

### Given:
The function is \( \sin(y) = 2x^3 - 2 \), and we need to find:

1. **(a)** \( \frac{dy}{dx} \) using implicit differentiation.
2. **(b)** The slope of the curve at the point \( (1, \pi) \).

#### Step (a): Use implicit differentiation to find \( \frac{dy}{dx} \).

Starting with the given equation:
\[
\sin(y) = 2x^3 - 2
\]

Differentiate both sides with respect to \( x \), keeping in mind that \( y \) is a function of \( x \):

\[
\frac{d}{dx}[\sin(y)] = \frac{d}{dx}[2x^3 - 2]
\]

- The derivative of \( \sin(y) \) with respect to \( x \) is \( \cos(y) \frac{dy}{dx} \), due to the chain rule.
- The derivative of \( 2x^3 - 2 \) with respect to \( x \) is \( 6x^2 \).

Thus, the equation becomes:
\[
\cos(y) \frac{dy}{dx} = 6x^2
\]

Solving for \( \frac{dy}{dx} \), we get:
\[
\frac{dy}{dx} = \frac{6x^2}{\cos(y)}
\]

#### Step (b): Find the slope at \( (1, \pi) \).

At the point \( (1, \pi) \), substitute \( x = 1 \) and \( y = \pi \).

- For \( x = 1 \), we have \( 6x^2 = 6(1)^2 = 6 \).
- For \( y = \pi \), \( \cos(\pi) = -1 \).

Now, substitute these values into the expression for \( \frac{dy}{dx} \):

\[
\frac{dy}{dx} = \frac{6}{-1} = -6
\]

So, the slope of the curve at the point \( (1, \pi) \) is \( -6 \).

---

Would you like a further breakdown of these steps? Here are some related questions:

1. How does the chain rule work in implicit differentiation?
2. Why is the slope negative at the point \( (1, \pi) \)?
3. What other types of functions require implicit differentiation?
4. Can you differentiate \( \sin(y) \) directly with respect to \( x \)?
5. How would the result change if we used another point?

**Tip:** Implicit differentiation is essential when you have equations involving both \( x \) and \( y \) without explicitly solving for one variable.

Use implicit differentiation to find dy/dx. Find the slope of the curve at the given point for sin(y) = 2x^3 - 2; (1, π).

Learn how to solve the equation sin(y) = 2x^3 - 2 using implicit differentiation. This problem covers trigonometric derivatives, chain rule application, and finding the slope at a specific point (1, π). Suitable for students in Grades 11-12.

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