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Sure, let's find the 20th derivative of the function \( f(x) = \sin(2x) \).

The derivatives of \( \sin(2x) \) follow a cyclical pattern. Let's explore this pattern:

1. **First Derivative:**
   \[
   f(x) = \sin(2x)
   \]
   The derivative of \( \sin(2x) \) is \( 2 \cos(2x) \).

   So, \( f'(x) = 2 \cos(2x) \).

2. **Second Derivative:**
   \[
   f'(x) = 2 \cos(2x)
   \]
   The derivative of \( 2 \cos(2x) \) is \( -4 \sin(2x) \).

   So, \( f''(x) = -4 \sin(2x) \).

3. **Third Derivative:**
   \[
   f''(x) = -4 \sin(2x)
   \]
   The derivative of \( -4 \sin(2x) \) is \( -8 \cos(2x) \).

   So, \( f'''(x) = -8 \cos(2x) \).

4. **Fourth Derivative:**
   \[
   f'''(x) = -8 \cos(2x)
   \]
   The derivative of \( -8 \cos(2x) \) is \( 16 \sin(2x) \).

   So, \( f^{(4)}(x) = 16 \sin(2x) \).

We see that every four derivatives, the function repeats with an additional multiplication factor of \( 2^n \).

So, the pattern of derivatives looks like this:
\[
\begin{aligned}
f(x) &= \sin(2x) \\
f'(x) &= 2 \cos(2x) \\
f''(x) &= -4 \sin(2x) \\
f'''(x) &= -8 \cos(2x) \\
f^{(4)}(x) &= 16 \sin(2x) \\
f^{(5)}(x) &= 32 \cos(2x) \\
f^{(6)}(x) &= -64 \sin(2x) \\
f^{(7)}(x) &= -128 \cos(2x) \\
f^{(8)}(x) &= 256 \sin(2x) \\
&\vdots \\
f^{(4n)}(x) &= 2^{2n} \sin(2x) \\
f^{(4n+1)}(x) &= 2^{2n+1} \cos(2x) \\
f^{(4n+2)}(x) &= -2^{2n+2} \sin(2x) \\
f^{(4n+3)}(x) &= -2^{2n+3} \cos(2x)
\end{aligned}
\]

For \( n = 5 \):
\[
\begin{aligned}
f^{(20)}(x) &= 2^{10} \sin(2x) \\
&= 1024 \sin(2x)
\end{aligned}
\]

So, the 20th derivative of \( f(x) = \sin(2x) \) is:
\[
f^{(20)}(x) = 1024 \sin(2x)
\]

Do you have any questions or want more details on any specific step?

Here are 8 relative questions to consider:
1. What is the general formula for the derivative of a trigonometric function?
2. How does the periodicity of trigonometric functions affect their higher-order derivatives?
3. What is the pattern of the derivatives of \( \cos(2x) \)?
4. How can you use Euler's formula to derive the derivatives of trigonometric functions?
5. What are the applications of higher-order derivatives in physics?
6. How can higher-order derivatives be used to solve differential equations?
7. What is the significance of the multiplication factor \( 2^n \) in the derivatives of \( \sin(2x) \)?
8. How does the chain rule apply to the differentiation of composite trigonometric functions?

**Tip:** Recognizing patterns in derivatives, especially for trigonometric and exponential functions, can greatly simplify the process of finding higher-order derivatives.

Discover how to find the 20th derivative of sin(2x) step-by-step. Learn about the cyclical pattern in derivatives of trigonometric functions and how to apply it to find higher-order derivatives effectively. Suitable for undergraduate students studying calculus.

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