derivative of `e^(x^2+1)`

Answer

To find the derivative of the function \( e^{(x^2 + 1)} \), we will use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the function
We start with the function:
\[ y = e^{(x^2 + 1)} \]

### Step 2: Apply the chain rule
According to the chain rule, if you have a function of the form \( e^{u} \), where \( u = x^2 + 1 \), the derivative is given by:
\[ \frac{dy}{dx} = e^{u} \cdot \frac{du}{dx} \]

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

### Step 4: Differentiate the inner function
Next, we need to find \( \frac{du}{dx} \) where \( u = x^2 + 1 \). The derivative of \( u \) is:
\[ \frac{du}{dx} = \frac{d}{dx}(x^2 + 1) = 2x + 0 = 2x \]
(Note: The derivative of a constant is 0.)

### Step 5: Combine the results
Now, substituting back into the chain rule formula:
\[ \frac{dy}{dx} = e^{(x^2 + 1)} \cdot (2x) \]

### Step 6: Write the final answer
Thus, the derivative of \( e^{(x^2 + 1)} \) is:
\[ \frac{dy}{dx} = 2x \cdot e^{(x^2 + 1)} \]

### Summary
The derivative of \( e^{(x^2 + 1)} \) is:
\[ \frac{dy}{dx} = 2x \cdot e^{(x^2 + 1)} \]

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