To differentiate the given function:

$y = 100e^{-1} \sin(x) + 800$

Step-by-Step Solution:

1. Constant Multiplier Rule: The term $100e^{-1}$ is a constant multiplier.

   • Differentiating $100e^{-1} \sin(x)$, we use the product of a constant and a function rule: [ \frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}[f(x)]. ]
   • Here, the derivative of $\sin(x)$ is $\cos(x)$.

   $\frac{d}{dx}[100e^{-1} \sin(x)] = 100e^{-1} \cos(x).$

2. Derivative of a Constant: The derivative of $800$ (a constant) is $0$.

Final Answer:

$\frac{dy}{dx} = 100e^{-1} \cos(x)$

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Do you have any questions or need further details?

Related Questions:

1. How would you differentiate $y = 100e^{-1} \cos(x) + 800$?
2. What is the derivative of $y = e^{-1} \sin(x) + e^{-1} \cos(x)$?
3. Can you explain the rules of differentiation for trigonometric functions?
4. How would the answer change if $y = 100e^{-x} \sin(x) + 800$?
5. What is the integral of $100e^{-1} \cos(x)$?

Tip:

Always handle constants (like $e^{-1}$) as fixed values during differentiation. This simplifies the problem-solving process!