y(x)=e^−1.35xsin(−1.25x) find derivatve

To find the derivative of the function \( y(x) = e^{-1.35x} \sin(-1.25x) \), we need to apply the product rule and the chain rule.

Given:
\[
y(x) = e^{-1.35x} \sin(-1.25x)
\]

### Step 1: Apply the Product Rule
The product rule states:
\[
\frac{d}{dx}[u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]
Here, let:
\[
u(x) = e^{-1.35x} \quad \text{and} \quad v(x) = \sin(-1.25x)
\]

### Step 2: Find the derivatives of \( u(x) \) and \( v(x) \)
- The derivative of \( u(x) = e^{-1.35x} \) with respect to \( x \) is:
  \[
  u'(x) = \frac{d}{dx} e^{-1.35x} = -1.35 e^{-1.35x}
  \]

- The derivative of \( v(x) = \sin(-1.25x) \) with respect to \( x \) is:
  \[
  v'(x) = \frac{d}{dx} \sin(-1.25x) = -1.25 \cos(-1.25x)
  \]

### Step 3: Combine using the Product Rule
Now, substitute the derivatives into the product rule:
\[
y'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]
Substitute the expressions for \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \):
\[
y'(x) = (-1.35 e^{-1.35x}) \cdot \sin(-1.25x) + e^{-1.35x} \cdot (-1.25 \cos(-1.25x))
\]

### Step 4: Simplify the Expression
\[
y'(x) = -1.35 e^{-1.35x} \sin(-1.25x) - 1.25 e^{-1.35x} \cos(-1.25x)
\]

So, the derivative is:
\[
y'(x) = -e^{-1.35x} \left(1.35 \sin(-1.25x) + 1.25 \cos(-1.25x)\right)
\]

Would you like further details on any of the steps or concepts used? 

### Related Questions:
1. How do you apply the chain rule to more complex functions?
2. What are the rules for differentiating trigonometric functions?
3. Can you explain how the exponential function's derivative works?
4. How does the sign of the sine or cosine function affect the derivative?
5. How would you differentiate a product of three or more functions?

### Tip:
Always double-check the signs when applying trigonometric derivatives, as they can change the outcome significantly.

Learn how to find the derivative of the function y(x) = e^(-1.35x) * sin(-1.25x) step-by-step using the product rule and chain rule. Understand the application of exponential functions and trigonometric functions in calculus.

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