Lecture on Using Chain Rule to Find nth Derivative

Objective

• Learn how to use the chain rule to find the nth derivative of functions.
• Specifically, focus on examples: finding the 100th derivative and finding general patterns.

Example 1: Function y = e^(-2x)

Derivatives

1. First Derivative (y')
   • Derivative of e^(-2x) = e^(-2x) * (-2)
   • Therefore, y' = -2e^(-2x)
2. Second Derivative (y'')
   • Keep -2, multiply by derivative of e^(-2x): -2 * e^(-2x) * (-2)
   • Therefore, y'' = (-2)^2 * e^(-2x)
3. Third Derivative (y''')
   • Keep (-2)^2, multiply by derivative of e^(-2x): (-2)^2 * e^(-2x) * (-2)
   • Therefore, y''' = (-2)^3 * e^(-2x)

Pattern

• nth derivative: y^(n) = (-2)^n * e^(-2x)
• Since (-2)^100 involves an even power, it can be written simply as 2^100.

Example 2: Function y = sin(3x)

Derivatives

1. First Derivative (y')
   • Derivative of sin(3x) = cos(3x) * 3
   • Therefore, y' = 3cos(3x)
2. Second Derivative (y'')
   • Derivative of 3cos(3x) = 3(-sin(3x) * 3)
   • Therefore, y'' = -3^2 * sin(3x)
3. Third Derivative (y''')
   • Derivative of -3^2 * sin(3x) = -3^2 * cos(3x) * 3
   • Therefore, y''' = -3^3 * cos(3x)
4. Fourth Derivative (y'''')
   • Derivative of -3^3 * cos(3x) = -3^3 * (-sin(3x) * 3)
   • Therefore, y'''' = 3^4 * sin(3x)

Pattern

• Noted behavior: Every 4th derivative results in the original function multiplied by a power of 3.
• For the 99th derivative: (-1)^(99/4) * 3^99 * cos(3x)

Example 3: Function y = 2^x

Derivatives

1. First Derivative (y')
   • Derivative of 2^x = 2^x * ln(2)
2. Second Derivative (y'')
   • 2^x * (ln(2))^2
3. Third Derivative (y''')
   • 2^x * (ln(2))^3

Pattern

• nth derivative: d^n(y) = 2^x * (ln(2))^n
• 10th derivative: 2^x * (ln(2))^10

Example 4: Function y = x * e^(-x)

Derivatives

1. First Derivative (y')
   • Derivative = e^(-x) - x * e^(-x)
2. Second Derivative (y'')
   • = -e^(-x) + (x - 2)e^(-x)

Pattern

• Alternating signs with coefficients forming an arithmetic sequence.
• 100th derivative: -100e^(-x) + x * e^(-x)

Example 5: Function y = sin^2(x) with 100th derivative

First Step

• Begin with rewriting the function: 2*sin(x)*cos(x) = sin(2x)
• Integrate the double angle identity.

Pattern

• Every 4 derivatives return the function to its original state (multiplied by 2^n as per the theorem).
• 99th derivative: (-1)^99 * 2^99 * cos(2x)

Conclusion

• Recognize patterns in derivatives to simplify repeated differentiation.
• Use trigonometric identities for more complex functions, power rules, and chain rules effectively.