find an equation of the tangent line to the curve y at the point. y = 4x sin(x) , ($$ \frac{\pi}{2},\text{ }2\pi) $$

To find the equation of the tangent line to the curve $ y = 4x \sin(x) $ at the point $ \left( \frac{\pi}{2}, 2\pi \right) $, we need two things:

1. The derivative of the function $ y $, which gives the slope of the tangent line.
2. The point on the curve, which we already have as $ \left( \frac{\pi}{2}, 2\pi \right) $.

### Step 1: Find the derivative of $ y = 4x \sin(x) $
To differentiate $ y = 4x \sin(x) $, we apply the product rule:

\[
\frac{d}{dx}[4x \sin(x)] = \frac{d}{dx}[4x] \cdot \sin(x) + 4x \cdot \frac{d}{dx}[\sin(x)]
\]

The derivative of $ 4x $ is 4, and the derivative of $ \sin(x) $ is $ \cos(x) $. So:

\[
\frac{dy}{dx} = 4 \sin(x) + 4x \cos(x)
\]

### Step 2: Evaluate the derivative at $ x = \frac{\pi}{2} $
We substitute $ x = \frac{\pi}{2} $ into the derivative to find the slope of the tangent line at that point.

\[
\frac{dy}{dx} \Bigg|_{x = \frac{\pi}{2}} = 4 \sin\left( \frac{\pi}{2} \right) + 4 \cdot \frac{\pi}{2} \cdot \cos\left( \frac{\pi}{2} \right)
\]

Since $ \sin\left( \frac{\pi}{2} \right) = 1 $ and $ \cos\left( \frac{\pi}{2} \right) = 0 $, we get:

\[
\frac{dy}{dx} \Bigg|_{x = \frac{\pi}{2}} = 4 \cdot 1 + 4 \cdot \frac{\pi}{2} \cdot 0 = 4
\]

So, the slope of the tangent line is $ m = 4 $.

### Step 3: Use the point-slope form of the line equation
The point-slope form of the equation of a line is:

\[
y - y_1 = m(x - x_1)
\]

Where $ m $ is the slope and $ (x_1, y_1) $ is the point on the line. Substituting $ m = 4 $, $ x_1 = \frac{\pi}{2} $, and $ y_1 = 2\pi $:

\[
y - 2\pi = 4\left(x - \frac{\pi}{2}\right)
\]

### Step 4: Simplify the equation
Now, we simplify the equation:

\[
y - 2\pi = 4x - 2\pi
\]

Adding $ 2\pi $ to both sides:

\[
y = 4x
\]

Thus, the equation of the tangent line is:

\[
y = 4x
\]

### Summary:
The equation of the tangent line to the curve $ y = 4x \sin(x) $ at the point $ \left( \frac{\pi}{2}, 2\pi \right) $ is $ y = 4x $.

Would you like more details on any of the steps? Or do you have other questions?

### 5 Follow-up Questions:
1. How do you differentiate products of functions (product rule)?
2. What does the point-slope form of a line equation look like?
3. How can I use the derivative to determine if a function is increasing or decreasing?
4. How do I evaluate trigonometric functions like $ \sin(x) $ and $ \cos(x) $ at key points?
5. How can I find the equation of the tangent line for other functions?

**Tip:** When using the product rule, remember that it applies when you have two functions multiplied together. The rule is: $ \frac{d}{dx}[uv] = u'v + uv' $, where $ u $ and $ v $ are functions of $ x $.

find an equation of the tangent line to the curve y at the point. y = 4x sin(x) , (π/2, 2π)

Learn how to find the equation of the tangent line to the curve y = 4x sin(x) at the point (π/2, 2π). This solution involves differentiation using the product rule and applying the point-slope form of a line. Suitable for advanced high school students studying calculus.

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