Questions: y=e^(a x^2), y''

Transcript text: $y=e^{a x^{2}}, y^{\prime \prime}$

Solution

To find the second derivative of the function $ y = e^{a x^2} $, we need to follow these steps:

Step 1: Define the Function

We start with the function given by

\[ y = e^{a x^2} \]

Step 2: Compute the First Derivative

Using the chain rule, we find the first derivative $ y' $:

\[ y' = \frac{d}{dx}(e^{a x^2}) = 2a x e^{a x^2} \]

Step 3: Compute the Second Derivative

Next, we differentiate $ y' $ to find the second derivative $ y'' $:

\[ y'' = \frac{d}{dx}(2a x e^{a x^2}) = 4a^2 x^2 e^{a x^2} + 2a e^{a x^2} \]

The second derivative of the function $ y = e^{a x^2} $ is

\[ \boxed{y'' = 4a^2 x^2 e^{a x^2} + 2a e^{a x^2}} \]

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