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To solve problem 13 (d), we need to expand the function \( f(x) = \sinh x \cdot \sinh 5x \) up to the order \( o(x^n) \) using the Maclaurin series.

Let’s start by breaking it down:

### Step 1: Maclaurin Series for \( \sinh x \)
The Maclaurin series expansion for the hyperbolic sine function \( \sinh x \) is:
\[
\sinh x = x + \frac{x^3}{6} + \frac{x^5}{120} + o(x^5)
\]

### Step 2: Apply the Series to \( \sinh 5x \)
Using the same series expansion for \( \sinh 5x \), we get:
\[
\sinh 5x = 5x + \frac{(5x)^3}{6} + \frac{(5x)^5}{120} + o(x^5)
= 5x + \frac{125x^3}{6} + \frac{3125x^5}{120} + o(x^5)
\]

### Step 3: Multiply the Expansions
Now, multiply \( \sinh x \) and \( \sinh 5x \):
\[
\sinh x \cdot \sinh 5x = \left( x + \frac{x^3}{6} + o(x^5) \right) \cdot \left( 5x + \frac{125x^3}{6} + o(x^5) \right)
\]

Expanding and keeping terms up to \( o(x^5) \):
1. The \( x \cdot 5x \) term gives \( 5x^2 \).
2. The \( x \cdot \frac{125x^3}{6} \) term gives \( \frac{125x^4}{6} \).
3. The \( \frac{x^3}{6} \cdot 5x \) term gives \( \frac{5x^4}{6} \).

Adding these terms:
\[
\sinh x \cdot \sinh 5x = 5x^2 + \frac{130x^4}{6} + o(x^5)
\]

### Final Answer:
\[
\sinh x \cdot \sinh 5x = 5x^2 + \frac{130x^4}{6} + o(x^5)
\]

Would you like further steps or explanations? 

### Related Questions:
1. What is the Maclaurin series for \( \cosh x \) up to the \( x^4 \) term?
2. How would the expansion change if we had \( \cosh x \cdot \sinh 5x \) instead?
3. Can you use the series expansion of \( e^x \) to derive \( \sinh x \) and \( \cosh x \) formulas?
4. How do Maclaurin and Taylor series differ when centered at different points?
5. What would be the result if \( f(x) = \sinh x + \sinh 5x \) was expanded instead?

### Tip:
When expanding products of functions in series, only keep terms that contribute to the order needed, as higher-order terms become negligible.

Expand sinh x * sinh 5x up to the order o(x^n) using the Maclaurin series. This problem involves key calculus concepts such as series expansion, hyperbolic functions, and Maclaurin series. Ideal for university students studying calculus or advanced mathematical series.

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