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Let's find the derivatives for each of these functions using the appropriate differentiation rules, such as the chain rule and quotient rule.

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### Problem 61
Given:
\[
y = (7x + 3)^4
\]

To differentiate, use the **chain rule**:
1. Let \( u = 7x + 3 \), then \( y = u^4 \).
2. The derivative of \( y \) with respect to \( u \) is \( \frac{dy}{du} = 4u^3 \).
3. The derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = 7 \).

Using the chain rule:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 4(7x + 3)^3 \cdot 7 = 28(7x + 3)^3
\]

---

### Problem 62
Given:
\[
y = (x^2 - 6)^3
\]

Use the **chain rule** here as well:
1. Let \( u = x^2 - 6 \), then \( y = u^3 \).
2. The derivative of \( y \) with respect to \( u \) is \( \frac{dy}{du} = 3u^2 \).
3. The derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = 2x \).

Using the chain rule:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 3(x^2 - 6)^2 \cdot 2x = 6x(x^2 - 6)^2
\]

---

### Problem 63
Given:
\[
y = \frac{1}{(x^2 + 5)^3}
\]

Rewrite the function as:
\[
y = (x^2 + 5)^{-3}
\]

Differentiate using the **chain rule**:
1. Let \( u = x^2 + 5 \), then \( y = u^{-3} \).
2. The derivative of \( y \) with respect to \( u \) is \( \frac{dy}{du} = -3u^{-4} \).
3. The derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = 2x \).

Using the chain rule:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -3(x^2 + 5)^{-4} \cdot 2x = -\frac{6x}{(x^2 + 5)^4}
\]

---

### Problem 64
Given:
\[
f(x) = \frac{1}{(5x + 1)^2}
\]

Rewrite the function as:
\[
f(x) = (5x + 1)^{-2}
\]

Differentiate using the **chain rule**:
1. Let \( u = 5x + 1 \), then \( f(x) = u^{-2} \).
2. The derivative of \( f \) with respect to \( u \) is \( \frac{df}{du} = -2u^{-3} \).
3. The derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = 5 \).

Using the chain rule:
\[
\frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} = -2(5x + 1)^{-3} \cdot 5 = -\frac{10}{(5x + 1)^3}
\]

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Would you like further details or explanations on any specific step?

### Related Questions:
1. How can the product rule be combined with the chain rule in differentiation?
2. What are the applications of derivatives in real-world problems?
3. How does implicit differentiation differ from the chain rule?
4. Can you differentiate composite functions that involve trigonometric terms?
5. How do we handle higher-order derivatives for similar types of functions?

### Tip:
Always simplify the expression of the derivative as much as possible to make it easier for further calculations or applications.

In Exercises 61–72, find the derivative of the function.

Step-by-step solutions for finding derivatives in calculus exercises. Learn how to apply the chain rule and quotient rule to differentiate functions, with examples like (7x + 3)^4 and 1/(5x + 1)^2. Suitable for Grade 12 and college-level calculus students.

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