Use the complex exponential to calculate
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To calculate the 26th derivative of the given expression using the complex exponential form, we'll express the hyperbolic and trigonometric functions in terms of exponentials. This approach leverages the properties of exponential functions to simplify differentiation.

### Given Expression:
\[
\frac{d^{26}}{dt^{26}} \left( \cosh(2t) \sin(2\sqrt{3}t) \right)
\]

### Step 1: Express the functions using complex exponentials

1. **Hyperbolic cosine**:
   \[
   \cosh(2t) = \frac{e^{2t} + e^{-2t}}{2}
   \]

2. **Sine function**:
   \[
   \sin(2\sqrt{3}t) = \frac{e^{i2\sqrt{3}t} - e^{-i2\sqrt{3}t}}{2i}
   \]

### Step 2: Combine the expressions

The product \( \cosh(2t) \sin(2\sqrt{3}t) \) becomes:
\[
\cosh(2t) \sin(2\sqrt{3}t) = \frac{e^{2t} + e^{-2t}}{2} \cdot \frac{e^{i2\sqrt{3}t} - e^{-i2\sqrt{3}t}}{2i}
\]

Expanding this:
\[
\cosh(2t) \sin(2\sqrt{3}t) = \frac{1}{4i} \left[ \left(e^{(2 + i2\sqrt{3})t} - e^{(2 - i2\sqrt{3})t}\right) + \left(e^{(-2 + i2\sqrt{3})t} - e^{(-2 - i2\sqrt{3})t}\right) \right]
\]

Simplify further:
\[
\cosh(2t) \sin(2\sqrt{3}t) = \frac{1}{4i} \left(e^{(2 + i2\sqrt{3})t} - e^{(2 - i2\sqrt{3})t} + e^{(-2 + i2\sqrt{3})t} - e^{(-2 - i2\sqrt{3})t}\right)
\]

### Step 3: Differentiate 26 times

To differentiate \( e^{\alpha t} \) with respect to \( t \) 26 times:
\[
\frac{d^{26}}{dt^{26}} e^{\alpha t} = \alpha^{26} e^{\alpha t}
\]

Apply this to each term in the expanded expression:
\[
\frac{d^{26}}{dt^{26}} \cosh(2t) \sin(2\sqrt{3}t) = \frac{1}{4i} \left[ (2 + i2\sqrt{3})^{26} e^{(2 + i2\sqrt{3})t} - (2 - i2\sqrt{3})^{26} e^{(2 - i2\sqrt{3})t} + (-2 + i2\sqrt{3})^{26} e^{(-2 + i2\sqrt{3})t} - (-2 - i2\sqrt{3})^{26} e^{(-2 - i2\sqrt{3})t} \right]
\]

### Step 4: Simplify and interpret the result

This result gives the 26th derivative as a linear combination of exponential functions, each with a coefficient that depends on a power of a complex number. In practice, these complex coefficients can be expressed in trigonometric form (using Euler's formula) to match the original sine and cosine components, if necessary.

### Final Answer:

The 26th derivative of \( \cosh(2t) \sin(2\sqrt{3}t) \) is:
\[
\frac{d^{26}}{dt^{26}} \left( \cosh(2t) \sin(2\sqrt{3}t) \right) = \frac{1}{4i} \left[ (2 + i2\sqrt{3})^{26} e^{(2 + i2\sqrt{3})t} - (2 - i2\sqrt{3})^{26} e^{(2 - i2\sqrt{3})t} + (-2 + i2\sqrt{3})^{26} e^{(-2 + i2\sqrt{3})t} - (-2 - i2\sqrt{3})^{26} e^{(-2 - i2\sqrt{3})t} \right]
\]

This expression represents the sum of four complex exponential terms. These terms might be further simplified or combined depending on the context or specific requirements.

Would you like to dive deeper into a particular aspect of this calculation, or do you have any further questions?

Here are some related questions you might explore:

1. How does the method of using complex exponentials simplify high-order derivatives?
2. What are the real and imaginary components of the complex coefficients?
3. How can Euler's formula be used to revert the result back to sine and cosine form?
4. What would the 26th derivative look like for other combinations of hyperbolic and trigonometric functions?
5. How does the approach change if the powers involved are even or odd?

**Tip:** Using the complex exponential form often simplifies the differentiation of trigonometric and hyperbolic functions, especially for higher-order derivatives.

Learn how to calculate the 26th derivative of a complex expression involving hyperbolic and trigonometric functions using complex exponentials. Step-by-step solution provided.

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