Find the 2nd dervative of `e^(ax+b)` with respect to x.

Answer

To find the second derivative of \( e^{ax + b} \) with respect to \( x \), we will follow these steps:

### Step 1: Define the function
Let:
\[ y = e^{ax + b} \]

### Step 2: Find the first derivative
To find the first derivative \( \frac{dy}{dx} \), we will use the chain rule. The derivative of \( e^u \) with respect to \( u \) is \( e^u \), and we need to multiply by the derivative of the exponent \( ax + b \).

1. Differentiate \( e^{ax + b} \):
\[ \frac{dy}{dx} = e^{ax + b} \cdot \frac{d}{dx}(ax + b) \]
2. The derivative of \( ax + b \) is \( a \) (since the derivative of \( b \) is 0):
\[ \frac{dy}{dx} = e^{ax + b} \cdot a \]
3. Thus, we can write:
\[ \frac{dy}{dx} = a e^{ax + b} \]

### Step 3: Find the second derivative
Now, we will differentiate \( \frac{dy}{dx} \) to find the second derivative \( \frac{d^2y}{dx^2} \).

1. Differentiate \( a e^{ax + b} \):
\[ \frac{d^2y}{dx^2} = a \cdot \frac{d}{dx}(e^{ax + b}) \]
2. Using the chain rule again:
\[ \frac{d}{dx}(e^{ax + b}) = e^{ax + b} \cdot \frac{d}{dx}(ax + b) = e^{ax + b} \cdot a \]
3. Therefore:
\[ \frac{d^2y}{dx^2} = a \cdot (a e^{ax + b}) = a^2 e^{ax + b} \]

### Final Result
The second derivative of \( e^{ax + b} \) with respect to \( x \) is:
\[ \frac{d^2y}{dx^2} = a^2 e^{ax + b} \]

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