Differentiate the following function:  f(x)=sinh(4x^(2)+5)

To differentiate the function \( f(x) = \sinh(4x^2 + 5) \), we will use the chain rule. The derivative of the hyperbolic sine function, \( \sinh(u) \), is \( \cosh(u) \), and we will also need to differentiate the inner function \( u = 4x^2 + 5 \).

1. **Differentiate the outer function**: 
   \[
   \frac{d}{dx} \sinh(u) = \cosh(u) \cdot \frac{du}{dx}
   \]

2. **Differentiate the inner function**:
   \[
   u = 4x^2 + 5 \quad \Rightarrow \quad \frac{du}{dx} = 8x
   \]

3. **Combine the results**:
   \[
   f'(x) = \cosh(4x^2 + 5) \cdot 8x
   \]

Thus, the derivative of the function is:
\[
f'(x) = 8x \cosh(4x^2 + 5)
\]

### Summary:
The derivative \( f'(x) \) is \( 8x \cosh(4x^2 + 5) \).

Do you want further details on the differentiation process or any other related concepts? Here are some questions you might find interesting:

1. What are the properties of hyperbolic functions?
2. How does the chain rule apply to other functions?
3. Can you explain the relationship between hyperbolic and trigonometric functions?
4. What are some applications of hyperbolic functions in real-world problems?
5. How would you integrate hyperbolic functions? 

**Tip**: When differentiating functions composed of other functions, always check for opportunities to apply the chain rule effectively.

Differentiate the following function: f(x)=sinh(4x^(2)+5)

Explore the differentiation of the function f(x) = sinh(4x^2 + 5) using the chain rule. This solution highlights key concepts in calculus, including the derivatives of hyperbolic functions. Suitable for students in Grades 11-12.

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