Find `dy/dx|_(x=pi/2)`, where `y=e^(sinx)`

Options

A. 0
B. 1
C. `-1`
D. 2

Answer

To find \(\frac{dy}{dx}\) at \(x = \frac{\pi}{2}\) for the function \(y = e^{\sin x}\), we will follow these steps:

### Step 1: Differentiate the function
We start with the function:
\[ y = e^{\sin x} \]
To differentiate \(y\) with respect to \(x\), we will use the chain rule. The derivative of \(e^{u}\) with respect to \(x\) is \(e^{u} \cdot \frac{du}{dx}\), where \(u = \sin x\).

Thus, we have:
\[ \frac{dy}{dx} = e^{\sin x} \cdot \frac{d}{dx}(\sin x) \]

### Step 2: Differentiate \(\sin x\)
The derivative of \(\sin x\) is \(\cos x\). Therefore, we can substitute this into our derivative:
\[ \frac{dy}{dx} = e^{\sin x} \cdot \cos x \]

### Step 3: Evaluate at \(x = \frac{\pi}{2}\)
Now we need to evaluate \(\frac{dy}{dx}\) at \(x = \frac{\pi}{2}\):
\[ \frac{dy}{dx}\bigg|_{x = \frac{\pi}{2}} = e^{\sin\left(\frac{\pi}{2}\right)} \cdot \cos\left(\frac{\pi}{2}\right) \]

### Step 4: Calculate \(\sin\left(\frac{\pi}{2}\right)\) and \(\cos\left(\frac{\pi}{2}\right)\)
We know that:
\[ \sin\left(\frac{\pi}{2}\right) = 1 \quad \text{and} \quad \cos\left(\frac{\pi}{2}\right) = 0 \]

### Step 5: Substitute the values
Substituting these values into our derivative:
\[ \frac{dy}{dx}\bigg|_{x = \frac{\pi}{2}} = e^{1} \cdot 0 = e \cdot 0 = 0 \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) at \(x = \frac{\pi}{2}\) is:
\[ \frac{dy}{dx}\bigg|_{x = \frac{\pi}{2}} = 0 \]

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