Find `f'(x) if f(x)=e^({x^(2)})`

Answer

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

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

### Step 2: Apply the chain rule
The chain rule states that if you have a composite function \( f(g(x)) \), the derivative is given by:
\[ f'(x) = f'(g(x)) \cdot g'(x) \]
In our case, let \( g(x) = x^2 \). Then \( f(g) = e^g \).

### Step 3: Differentiate the outer function
The derivative of \( e^g \) with respect to \( g \) is:
\[ \frac{d}{dg}(e^g) = e^g \]

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

### Step 5: Combine the results
Using the chain rule, we combine the derivatives:
\[ f'(x) = e^{g(x)} \cdot g'(x) = e^{x^2} \cdot 2x \]

### Final result
Thus, the derivative of \( f(x) = e^{x^2} \) is:
\[ f'(x) = 2x e^{x^2} \]

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