We are tasked with finding the rate of change of the equation $\sin(y) + x = \frac{7}{2}$ at the point $(3, \frac{\pi}{6})$. This involves finding the total derivative, which can be interpreted as the rate of change of $y$ with respect to $x$.

Step 1: Implicit Differentiation

We differentiate both sides of the equation $\sin(y) + x = \frac{7}{2}$ with respect to $x$, treating $y$ as a function of $x$.

$\frac{d}{dx} \left( \sin(y) + x \right) = \frac{d}{dx} \left( \frac{7}{2} \right)$

Since $\frac{7}{2}$ is a constant, its derivative is 0. Now apply the derivative to each term:

• The derivative of $\sin(y)$ with respect to $x$ is $\cos(y) \cdot \frac{dy}{dx}$ by the chain rule.
• The derivative of $x$ with respect to $x$ is 1.

Thus, we get the equation:

$\cos(y) \cdot \frac{dy}{dx} + 1 = 0$

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

Now, solve for $\frac{dy}{dx}$ (the rate of change of $y$ with respect to $x$):

$\cos(y) \cdot \frac{dy}{dx} = -1$ $\frac{dy}{dx} = \frac{-1}{\cos(y)}$

Step 3: Substitute the Given Point

We are given the point $(3, \frac{\pi}{6})$, which means $x = 3$ and $y = \frac{\pi}{6}$. At this point:

$\cos\left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2}$

Now substitute this value into the equation for $\frac{dy}{dx}$:

$\frac{dy}{dx} = \frac{-1}{\frac{\sqrt{3}}{2}} = \frac{-2}{\sqrt{3}}$

Step 4: Simplify the Result

To simplify, multiply the numerator and denominator by $\sqrt{3}$:

$\frac{dy}{dx} = \frac{-2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{-2\sqrt{3}}{3}$

Thus, the rate of change of $y$ with respect to $x$ at the point $(3, \frac{\pi}{6})$ is:

$\frac{dy}{dx} = \frac{-2\sqrt{3}}{3}$

Do you need any further clarifications or details on this?

Related Questions:

1. How do we apply implicit differentiation to more complex functions?
2. What is the geometric interpretation of $\frac{dy}{dx}$ in terms of the slope of the curve?
3. How does chain rule apply when differentiating trigonometric functions?
4. Can we find the second derivative from the given equation?
5. How would the rate of change differ at another point on this curve?

Tip:

Always ensure you understand the chain rule when differentiating implicitly, as it is crucial for handling functions where $y$ is a function of $x$.