Find `(dy)/(dx)|_(x= pi/2)` where `y =e^("sin x")`.

Options

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

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 \(y\) with respect to \(x\)

Given:
\[ y = e^{\sin x} \]

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:
\[ \frac{d}{dx}(\sin x) = \cos x \]

### Step 3: Substitute back into the derivative

Now substituting back, we get:
\[ \frac{dy}{dx} = e^{\sin x} \cdot \cos x \]

### Step 4: Evaluate \(\frac{dy}{dx}\) 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 5: Calculate \(\sin\left(\frac{\pi}{2}\right)\) and \(\cos\left(\frac{\pi}{2}\right)\)

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

### Step 6: Substitute these values

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

### 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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