2xy+π sin y = 2π I want to derive it and then replace it with the point (x=1,y=90)

Let's first derive the given equation with respect to \( x \), and then substitute the point \((x = 1, y = 90^\circ)\) into the derived equation.

The equation provided is:

\[
2xy + \pi \sin(y) = 2\pi
\]

### Step 1: Implicit Differentiation

We will differentiate both sides of the equation with respect to \( x \), treating \( y \) as a function of \( x \) (i.e., \( y = y(x) \)).

1. **Differentiate the term \( 2xy \):**

\[
\frac{d}{dx}(2xy) = 2y + 2x \frac{dy}{dx}
\]

2. **Differentiate the term \( \pi \sin(y) \):**

\[
\frac{d}{dx}(\pi \sin(y)) = \pi \cos(y) \frac{dy}{dx}
\]

3. **Differentiate the constant term \( 2\pi \):**

\[
\frac{d}{dx}(2\pi) = 0
\]

Now, putting everything together:

\[
2y + 2x \frac{dy}{dx} + \pi \cos(y) \frac{dy}{dx} = 0
\]

### Step 2: Solve for \(\frac{dy}{dx}\)

Now, we'll solve this equation for \(\frac{dy}{dx}\):

\[
2x \frac{dy}{dx} + \pi \cos(y) \frac{dy}{dx} = -2y
\]

Factor out \(\frac{dy}{dx}\) on the left side:

\[
\frac{dy}{dx}(2x + \pi \cos(y)) = -2y
\]

Finally, solve for \(\frac{dy}{dx}\):

\[
\frac{dy}{dx} = \frac{-2y}{2x + \pi \cos(y)}
\]

### Step 3: Substitute \( x = 1 \) and \( y = 90^\circ \)

Now, substitute the values \( x = 1 \) and \( y = 90^\circ \) into the derived equation. Remember that:

\[
\cos(90^\circ) = 0
\]

So the equation becomes:

\[
\frac{dy}{dx} = \frac{-2(90)}{2(1) + \pi \cdot 0} = \frac{-180}{2} = -90
\]

Thus, at the point \( (x = 1, y = 90^\circ) \), the derivative \(\frac{dy}{dx}\) is \(-90\).

---

Would you like more details or have any questions about this solution?

### Related Questions:
1. How do you perform implicit differentiation in general?
2. What are the rules for differentiating trigonometric functions?
3. How does the chain rule apply in implicit differentiation?
4. How do you interpret the derivative in a real-world context?
5. How would the derivative change if \( y \) were in radians instead of degrees?
6. How do you differentiate a product of two functions?
7. What is the geometric interpretation of the derivative at a point?
8. How do you find the second derivative after finding the first derivative implicitly?

### Tip:
Always remember to check the units (degrees vs. radians) when working with trigonometric functions in calculus.

Learn how to derive and substitute in the equation 2xy + π sin y = 2π with a detailed step-by-step solution. This problem involves implicit differentiation and trigonometric functions. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation