# Let's calculate the derivative at x = -0.69 using the derived formula.
import math

# Given x value
x = -0.69

# Coefficients
a = 1.35
b = 1.25

# Calculate the derivative y'(x) at x = -0.69
y_prime = -math.exp(-a * x) * (a * math.sin(-b * x) + b * math.cos(-b * x))
y_primeThe derivative of the function \( y(x) = e^{-1.35x} \sin(-1.25x) \) at \( x = -0.69 \) is approximately \( y'(-0.69) \approx -4.667 \).

Do you have any further questions, or would you like more details on this calculation?

### Related Questions:
1. How would the derivative change if the coefficients in the function were different?
2. What is the significance of the negative sign in the derivative?
3. How can you find the second derivative of this function?
4. How do derivatives behave near points where the function crosses the x-axis?
5. How would you estimate the function's behavior near \( x = -0.69 \) using this derivative?

### Tip:
When substituting a value into a derivative, ensure the trigonometric function inputs are in radians, as is standard in calculus.

Learn how to calculate the derivative of the function y(x) = e^(-1.35x) * sin(-1.25x) at x = -0.69. This problem involves calculus concepts, including derivatives of exponential and trigonometric functions, and applies the chain rule. Suitable for college-level students.

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