Find the derivative of `y = e^(x^2)`

Answer

To find the derivative of the function \( y = e^{x^2} \), we will use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the function
We have the function:
\[ y = e^{x^2} \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \), we apply the chain rule. The chain rule states that if you have a composite function \( y = f(g(x)) \), then the derivative \( \frac{dy}{dx} \) is given by:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
In our case, \( f(u) = e^u \) where \( u = x^2 \).

### Step 3: Differentiate the outer function
The derivative of \( f(u) = e^u \) with respect to \( u \) is:
\[ f'(u) = e^u \]

### Step 4: Differentiate the inner function
Now we differentiate the inner function \( g(x) = x^2 \):
\[ g'(x) = \frac{d}{dx}(x^2) = 2x \]

### Step 5: Apply the chain rule
Now we can combine these results using the chain rule:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) = e^{x^2} \cdot 2x \]

### Step 6: Write the final answer
Thus, the derivative of \( y = e^{x^2} \) is:
\[ \frac{dy}{dx} = 2x e^{x^2} \]

### Final Answer:
\[ \frac{dy}{dx} = 2x e^{x^2} \]

---