The derivative of sin x with respect to `e ^(x)` is :

Options

A. `- e ^(x) cos x `
B. `- e ^(-x) cos x `
C. `e ^(-x) cos x `
D. `e ^(x) cos x `

Answer

To find the derivative of \( \sin x \) with respect to \( e^x \), we can follow these steps:

### Step 1: Define the functions
Let:
- \( f_1 = \sin x \)
- \( f_2 = e^x \)

### Step 2: Use the chain rule
We want to find \( \frac{d}{d(e^x)}(\sin x) \). By the chain rule, we have:
\[ \frac{d}{d(e^x)}(\sin x) = \frac{d(\sin x)}{dx} \cdot \frac{dx}{d(e^x)} \]

### Step 3: Calculate the derivatives
1. Calculate \( \frac{d(\sin x)}{dx} \):
\[ \frac{d(\sin x)}{dx} = \cos x \]

2. Calculate \( \frac{dx}{d(e^x)} \):
Since \( e^x \) is a function of \( x \), we can find its derivative:
\[ \frac{d(e^x)}{dx} = e^x \]
Therefore, the reciprocal is:
\[ \frac{dx}{d(e^x)} = \frac{1}{e^x} \]

### Step 4: Combine the results
Now substitute these derivatives back into the chain rule expression:
\[ \frac{d}{d(e^x)}(\sin x) = \cos x \cdot \frac{1}{e^x} = \frac{\cos x}{e^x} \]

### Step 5: Simplify the expression
We can rewrite the expression as:
\[ \frac{\cos x}{e^x} = \cos x \cdot e^{-x} \]

### Final Answer
Thus, the derivative of \( \sin x \) with respect to \( e^x \) is:
\[ \frac{\cos x}{e^x} \quad \text{or} \quad \cos x \cdot e^{-x} \]