Let `y=ax^(2)+3`, where 'a' is constant. Find `(dy)/(dx)` by the delta method and find `[(dy)/(dx)]_(x=-1)`.

Answer

To find the derivative of the function \( y = ax^2 + 3 \) using the delta method, we will follow these steps:

### Step 1: Define the function and the increment
We start with the function:
\[ y = ax^2 + 3 \]
We introduce a small increment \( \Delta x \) to \( x \), so we have:
\[ y + \Delta y = a(x + \Delta x)^2 + 3 \]

### Step 2: Expand the equation
Now we expand the right-hand side:
\[ y + \Delta y = a((x + \Delta x)^2) + 3 = a(x^2 + 2x\Delta x + (\Delta x)^2) + 3 \]
This simplifies to:
\[ y + \Delta y = ax^2 + 2ax\Delta x + a(\Delta x)^2 + 3 \]

### Step 3: Isolate \( \Delta y \)
Next, we isolate \( \Delta y \):
\[ \Delta y = (ax^2 + 2ax\Delta x + a(\Delta x)^2 + 3) - (ax^2 + 3) \]
This simplifies to:
\[ \Delta y = 2ax\Delta x + a(\Delta x)^2 \]

### Step 4: Divide by \( \Delta x \)
Now, we divide both sides by \( \Delta x \):
\[ \frac{\Delta y}{\Delta x} = 2ax + a\Delta x \]

### Step 5: Take the limit as \( \Delta x \) approaches 0
To find the derivative \( \frac{dy}{dx} \), we take the limit as \( \Delta x \) approaches 0:
\[ \frac{dy}{dx} = \lim_{\Delta x \to 0} \left( 2ax + a\Delta x \right) = 2ax + 0 = 2ax \]

### Step 6: Evaluate \( \frac{dy}{dx} \) at \( x = -1 \)
Now, we need to find \( \frac{dy}{dx} \) at \( x = -1 \):
\[ \frac{dy}{dx} \bigg|_{x = -1} = 2a(-1) = -2a \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) at \( x = -1 \) is:
\[ \frac{dy}{dx} \bigg|_{x = -1} = -2a \]

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