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This image contains an assignment on Higher Order Derivatives and Implicit Differentiation. Let's work through some of the problems and their solutions. I will guide you through a few questions here and provide explanations where needed.

### 1. Problem 1:
Find the 3rd derivative of \( y = x^4 + 5x^3 + x \)

- First derivative:
  \[
  y' = 4x^3 + 15x^2 + 1
  \]
- Second derivative:
  \[
  y'' = 12x^2 + 30x
  \]
- Third derivative:
  \[
  y''' = 24x + 30
  \]
Answer: **c. \( y''' = 24x + 30 \)**

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### 2. Problem 2:
Find the 2nd derivative of \( y = e^{5x} + \cos(2x) \)

- First derivative:
  \[
  y' = 5e^{5x} - 2\sin(2x)
  \]
- Second derivative:
  \[
  y'' = 25e^{5x} - 4\cos(2x)
  \]
Answer: **a. \( y'' = 25e^{5x} - 4\cos(2x) \)**

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### 3. Problem 3:
Find the 3rd derivative of \( y = (2x + 6)^6 \)

- Using the chain rule:
  - First derivative:
    \[
    y' = 12(2x + 6)^5
    \]
  - Second derivative:
    \[
    y'' = 120(2x + 6)^4
    \]
  - Third derivative:
    \[
    y''' = 960(2x + 6)^3
    \]
Answer: **c. \( y''' = 960(2x + 6)^3 \)**

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### 4. Problem 4:
Find the 1004th derivative of \( y = \sin(x) \)

The sine function cycles between sine and cosine every two derivatives:
\[
\sin(x) \rightarrow \cos(x) \rightarrow -\sin(x) \rightarrow -\cos(x) \rightarrow \sin(x) \quad (\text{and repeat})
\]
Thus, since 1004 mod 4 is 0, the answer is \( \sin(x) \).

Answer: **a. \( y^{(1004)} = \sin(x) \)**

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### 5. Problem 5:
Find the 14th derivative of \( y = 1/x \)

The derivative of \( 1/x \) is \( -1/x^2 \), and each subsequent derivative produces higher powers of \( x \) in the denominator. The 14th derivative is given by:
\[
y^{(14)} = \frac{14!}{x^{15}}
\]
Answer: **b. \( \frac{14!}{x^{15}} \)**

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These are the worked-out solutions for five of the problems. Would you like explanations or help with the remaining problems?

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Here are five related questions for further exploration:
1. How does the chain rule apply in calculating derivatives for composite functions?
2. What patterns can be observed in higher-order derivatives of sine and cosine functions?
3. Why do we see factorial terms in higher-order derivatives of functions like \( 1/x \)?
4. How can we generalize the nth derivative for functions of the form \( e^{kx} \)?
5. What is the significance of alternating signs in the derivatives of trigonometric functions?

**Tip**: For complex functions, breaking them down into simpler components and applying rules like the chain rule can greatly simplify the process of differentiation.

Assignment No. 6: Higher Order Derivatives and Implicit Differentiation.

Solve problems related to higher-order derivatives and implicit differentiation. This assignment covers exponential, trigonometric, and logarithmic derivatives, including topics like the chain rule and product rule, designed for advanced calculus learners.

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