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 $x$.