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### **30.** \( f(x) = e^x \cosh x \)

To find the derivative, use the product rule: 
\[
(fg)' = f'g + fg'.
\]

Here, \( f(x) = e^x \) and \( g(x) = \cosh x \). Their derivatives are:
\[
f'(x) = e^x, \quad g'(x) = \sinh x.
\]

Applying the product rule:
\[
f'(x) = e^x \cosh x + e^x \sinh x.
\]

This can be factored as:
\[
f'(x) = e^x (\cosh x + \sinh x).
\]

### **31.** \( f(x) = \tanh(\sqrt{x}) \)

To find this derivative, use the chain rule. Let:
\[
u = \sqrt{x}, \quad f(x) = \tanh(u).
\]

First, find the derivatives:
\[
\frac{du}{dx} = \frac{1}{2\sqrt{x}}, \quad \frac{df}{du} = \text{sech}^2(u).
\]

Now, using the chain rule:
\[
f'(x) = \frac{df}{du} \cdot \frac{du}{dx} = \text{sech}^2(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}.
\]

So,
\[
f'(x) = \frac{\text{sech}^2(\sqrt{x})}{2\sqrt{x}}.
\]

### **32.** \( g(x) = \sinh^2(x) \)

Here, use the chain rule. Let:
\[
u = \sinh(x), \quad g(x) = u^2.
\]

First, find the derivatives:
\[
\frac{du}{dx} = \cosh(x), \quad \frac{dg}{du} = 2u = 2\sinh(x).
\]

Now, using the chain rule:
\[
g'(x) = \frac{dg}{du} \cdot \frac{du}{dx} = 2\sinh(x) \cdot \cosh(x).
\]

This can be simplified using the double-angle identity for hyperbolic functions:
\[
g'(x) = \sinh(2x).
\]

---

### Summary of the Answers

1. **30.** \( f'(x) = e^x (\cosh x + \sinh x) \).
2. **31.** \( f'(x) = \frac{\text{sech}^2(\sqrt{x})}{2\sqrt{x}} \).
3. **32.** \( g'(x) = \sinh(2x) \).

Would you like more details or have any questions?

### 5 Related Questions

1. How do you use the product rule when differentiating products of more than two functions?
2. What is the chain rule, and why is it important in calculus?
3. Can you derive the derivatives of the hyperbolic functions \(\sinh x\) and \(\cosh x\) from their definitions?
4. How do you simplify expressions involving hyperbolic identities, such as \(\sinh(2x)\)?
5. What are some real-world applications of hyperbolic functions in calculus?

### Tip

When using the chain rule, always identify the inner and outer functions clearly to avoid mistakes in differentiation.

Find the derivative. Simplify where possible.
30. f(x) = e^x cosh(x)
31. f(x) = tanh(√x)
32. g(x) = sinh^2(x)

Find the derivatives of e^x cosh(x), tanh(√x), and sinh^2(x) using product rule and chain rule. This calculus problem covers key concepts like hyperbolic functions and their identities.

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