If ` y^(2) = ax^(2) + b , " then " (d^(2)y)/( dx^(2))`

Options

A. ` (ab)/x^(3) `
B. ` x^(3)/(ab)`
C. ` (ab)/y^(2)`
D. ` (ab)/(y^(3)`

Answer

To solve the problem where we need to find \(\frac{d^2y}{dx^2}\) given that \(y^2 = ax^2 + b\), we can follow these steps:

### Step 1: Differentiate \(y^2 = ax^2 + b\)
We start by differentiating both sides with respect to \(x\).

\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(ax^2 + b) \]

Using the chain rule on the left side and the power rule on the right side, we have:

\[ 2y \frac{dy}{dx} = 2ax \]

### Step 2: Solve for \(\frac{dy}{dx}\)
Now, we can isolate \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} = \frac{2ax}{2y} = \frac{ax}{y} \]

### Step 3: Differentiate \(\frac{dy}{dx}\) to find \(\frac{d^2y}{dx^2}\)
Next, we differentiate \(\frac{dy}{dx}\) again to find \(\frac{d^2y}{dx^2}\). We will use the quotient rule here, which states that if \(u = ax\) and \(v = y\), then:

\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

Applying this to \(\frac{dy}{dx} = \frac{ax}{y}\):

- \(u = ax\) and \(v = y\)
- \(\frac{du}{dx} = a\)
- \(\frac{dv}{dx} = \frac{dy}{dx} = \frac{ax}{y}\)

Now, we can apply the quotient rule:

\[ \frac{d^2y}{dx^2} = \frac{y(a) - ax\left(\frac{ax}{y}\right)}{y^2} \]

### Step 4: Simplify the expression
Substituting \(\frac{dy}{dx}\) into the equation:

\[ \frac{d^2y}{dx^2} = \frac{ay - \frac{a^2x^2}{y}}{y^2} \]

Combining the terms in the numerator:

\[ \frac{d^2y}{dx^2} = \frac{ay^2 - a^2x^2}{y^3} \]

### Step 5: Substitute \(y^2\) from the original equation
From the original equation \(y^2 = ax^2 + b\), we can substitute \(y^2\) into our expression:

\[ \frac{d^2y}{dx^2} = \frac{a(ax^2 + b) - a^2x^2}{(y^2)^{3/2}} \]

This simplifies to:

\[ \frac{d^2y}{dx^2} = \frac{ab}{y^3} \]

### Final Answer
Thus, the final result is:

\[ \frac{d^2y}{dx^2} = \frac{ab}{y^3} \]