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

Options

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

Answer

To find \(\frac{dy}{dx}\) at \(x = \frac{\pi}{2}\) where \(y = e^{\sin x}\), 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 \(u = \sin x\).

So,
\[ \frac{dy}{dx} = e^{\sin x} \cdot \frac{d}{dx}(\sin x) \]

### Step 2: Differentiate \(\sin x\)

The derivative of \(\sin x\) is \(\cos x\). Therefore,
\[ \frac{dy}{dx} = e^{\sin x} \cdot \cos x \]

### Step 3: Evaluate \(\frac{dy}{dx}\) at \(x = \frac{\pi}{2}\)

Now we substitute \(x = \frac{\pi}{2}\) into the derivative:
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{2}} = e^{\sin(\frac{\pi}{2})} \cdot \cos(\frac{\pi}{2}) \]

### Step 4: Calculate \(\sin(\frac{\pi}{2})\) and \(\cos(\frac{\pi}{2})\)

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

### Step 5: Substitute these values into the expression

Thus,
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{2}} = e^{1} \cdot 0 = e \cdot 0 = 0 \]

### Conclusion

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

---