To differentiate the function $f(x) = \frac{\sin(2x)}{\sin(x)}$, we will apply the quotient rule.

The quotient rule for differentiation is:

$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$

Here, $g(x) = \sin(2x)$ and $h(x) = \sin(x)$.

Step 1: Differentiate the numerator $g(x) = \sin(2x)$

$g'(x) = 2\cos(2x)$

Step 2: Differentiate the denominator $h(x) = \sin(x)$

$h'(x) = \cos(x)$

Step 3: Apply the quotient rule

$f'(x) = \frac{(2\cos(2x))\sin(x) - \sin(2x)\cos(x)}{(\sin(x))^2}$

Now we simplify the expression.

Step 4: Use the identity for $\sin(2x) = 2\sin(x)\cos(x)$

Substitute $\sin(2x)$ in the expression: $f'(x) = \frac{2\cos(2x)\sin(x) - 2\sin(x)\cos(x)\cos(x)}{\sin^2(x)}$

You can leave the derivative in this form or simplify further depending on the situation.

Would you like a more detailed explanation or have any further questions?

Here are some related questions:

1. How does the quotient rule work with trigonometric functions?
2. Can the derivative be simplified further?
3. What other trigonometric identities might simplify the result?
4. How does the chain rule apply in the differentiation of $\sin(2x)$?
5. What happens when $\sin(x) = 0$ in the function?

Tip: Trigonometric identities, like the double angle formulas, often simplify derivatives involving sine and cosine functions.