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. $y''' = 24x + 30$

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. $y'' = 25e^{5x} - 4\cos(2x)$

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. $y''' = 960(2x + 6)^3$

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. $y^{(1004)} = \sin(x)$

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. $\frac{14!}{x^{15}}$

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

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.