If `y=f(x^(2))andf'(x)=e^(sqrt(x))` then `(dy)/(dx)` is equal to

Options

A. `2xe^(2sqrt(x))`
B. `2x(e^x)`
C. `4xe^(sqrt(x))`
D. `4xe^(x)`

Answer

To solve the problem, we need to find the derivative \(\frac{dy}{dx}\) given that \(y = f(x^2)\) and \(f'(x) = e^{\sqrt{x}}\).

### Step-by-Step Solution:

1. **Identify the function and its derivative**:
We have \(y = f(x^2)\). To find \(\frac{dy}{dx}\), we will use the chain rule.

2. **Apply the chain rule**:
According to the chain rule, if \(y = f(u)\) where \(u = x^2\), then:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Here, \(u = x^2\), so we need to find \(\frac{du}{dx}\).

3. **Calculate \(\frac{du}{dx}\)**:
The derivative of \(u = x^2\) with respect to \(x\) is:
\[ \frac{du}{dx} = 2x \]

4. **Find \(\frac{dy}{du}\)**:
From the problem, we know that \(f'(x) = e^{\sqrt{x}}\). Therefore, we need to evaluate \(f'(u)\) where \(u = x^2\):
\[ \frac{dy}{du} = f'(x^2) = e^{\sqrt{x^2}} = e^{x} \]

5. **Combine the results**:
Now we can substitute \(\frac{dy}{du}\) and \(\frac{du}{dx}\) back into the chain rule expression:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = e^{x} \cdot 2x \]

6. **Final expression**:
Thus, we have:
\[ \frac{dy}{dx} = 2x e^{x} \]

### Conclusion:
The derivative \(\frac{dy}{dx}\) is equal to \(2x e^{x}\).