Using Calculus rules, determine the derivative of the following functions:
(a) y = √x cos(x2 + 1).
(b) y = ln( 3
pe5x + e−2x)

Let's determine the derivatives of the given functions using calculus rules:

### (a) \( y = \sqrt{x} \cos(x^2 + 1) \)

To differentiate this function, we apply both the **product rule** and the **chain rule**.

- First, rewrite \( \sqrt{x} \) as \( x^{1/2} \).
- Let \( u(x) = x^{1/2} \) and \( v(x) = \cos(x^2 + 1) \).

The product rule states:
\[
\frac{d}{dx}[u(x) v(x)] = u'(x) v(x) + u(x) v'(x)
\]

Now, differentiate each part:

1. \( u(x) = x^{1/2} \) implies \( u'(x) = \frac{1}{2}x^{-1/2} \).
2. \( v(x) = \cos(x^2 + 1) \) requires using the chain rule:
   - The derivative of \( \cos(x^2 + 1) \) is \( -\sin(x^2 + 1) \times \frac{d}{dx}(x^2 + 1) \).
   - The derivative of \( x^2 + 1 \) is \( 2x \).
   - Thus, \( v'(x) = -2x \sin(x^2 + 1) \).

Now, apply the product rule:

\[
\frac{dy}{dx} = \left(\frac{1}{2}x^{-1/2}\right) \cos(x^2 + 1) + x^{1/2} \left(-2x \sin(x^2 + 1)\right)
\]

Simplifying:

\[
\frac{dy}{dx} = \frac{1}{2\sqrt{x}} \cos(x^2 + 1) - 2x^{3/2} \sin(x^2 + 1)
\]

---

### (b) \( y = \ln \left( \frac{3}{\pi e^{5x} + e^{-2x}} \right) \)

For this, we first simplify the logarithmic expression and then apply the chain rule.

We can rewrite the function as:
\[
y = \ln(3) - \ln(\pi e^{5x} + e^{-2x})
\]

The derivative of \( y \) will involve two parts:

1. The derivative of \( \ln(3) \) is 0 (since \( \ln(3) \) is a constant).
2. Now, apply the chain rule to differentiate \( \ln(\pi e^{5x} + e^{-2x}) \):

Let \( u(x) = \pi e^{5x} + e^{-2x} \). The derivative of \( \ln(u(x)) \) is \( \frac{1}{u(x)} \times u'(x) \).

Now, find \( u'(x) \):

\[
u'(x) = \frac{d}{dx}(\pi e^{5x} + e^{-2x}) = \pi \cdot 5e^{5x} - 2e^{-2x}
\]

Thus, the derivative of \( y \) is:

\[
\frac{dy}{dx} = -\frac{\pi \cdot 5e^{5x} - 2e^{-2x}}{\pi e^{5x} + e^{-2x}}
\]

---

### Final Answers:
(a) \( \frac{dy}{dx} = \frac{1}{2\sqrt{x}} \cos(x^2 + 1) - 2x^{3/2} \sin(x^2 + 1) \)

(b) \( \frac{dy}{dx} = -\frac{\pi \cdot 5e^{5x} - 2e^{-2x}}{\pi e^{5x} + e^{-2x}} \)

Let me know if you'd like further details or clarifications.

### Related Questions:
1. How do you apply the chain rule to more complex functions?
2. What is the significance of the product rule in calculus?
3. How do you differentiate natural logarithmic functions involving sums?
4. Can you explain how the derivative of exponential functions works?
5. How does simplifying logarithmic expressions affect differentiation?

### Tip:
Always simplify the function as much as possible before applying differentiation rules. This reduces complexity and avoids mistakes.

Using Calculus rules, determine the derivative of the following functions: (a) y = √x cos(x² + 1). (b) y = ln(3 / (πe⁵ˣ + e⁻²ˣ))

Learn how to differentiate the functions y = √x cos(x² + 1) and y = ln(3 / (πe⁵ˣ + e⁻²ˣ)) using the product rule, chain rule, and logarithmic differentiation. This detailed step-by-step solution explains the application of key calculus rules for college-level students.

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