Find `(d^(2)y)/(dx^(2))` in the following
If `x^(2/3)+y^(2/3)=a^(2/3),"find "(d^(2)y)/(dx^(2))`

Answer

To find \(\frac{d^2y}{dx^2}\) given the equation \(x^{2/3} + y^{2/3} = a^{2/3}\), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation:
\[ x^{2/3} + y^{2/3} = a^{2/3} \]
Differentiating both sides with respect to \(x\):
\[ \frac{d}{dx}(x^{2/3}) + \frac{d}{dx}(y^{2/3}) = \frac{d}{dx}(a^{2/3}) \]
Since \(a\) is a constant, the derivative of \(a^{2/3}\) is 0. Thus, we have:
\[ \frac{2}{3}x^{-1/3} + \frac{2}{3}y^{-1/3} \frac{dy}{dx} = 0 \]

### Step 2: Solve for \(\frac{dy}{dx}\)
Rearranging the equation gives:
\[ \frac{2}{3}y^{-1/3} \frac{dy}{dx} = -\frac{2}{3}x^{-1/3} \]
Dividing both sides by \(\frac{2}{3}y^{-1/3}\):
\[ \frac{dy}{dx} = -\frac{y^{1/3}}{x^{1/3}} \]

### Step 3: Differentiate \(\frac{dy}{dx}\) to find \(\frac{d^2y}{dx^2}\)
Now we differentiate \(\frac{dy}{dx}\):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(-\frac{y^{1/3}}{x^{1/3}}\right) \]
Using the quotient rule:
\[ \frac{d^2y}{dx^2} = -\frac{(x^{1/3} \cdot \frac{d}{dx}(y^{1/3}) - y^{1/3} \cdot \frac{d}{dx}(x^{1/3})}{(x^{1/3})^2} \]

### Step 4: Calculate \(\frac{d}{dx}(y^{1/3})\) and \(\frac{d}{dx}(x^{1/3})\)
Using the chain rule:
\[ \frac{d}{dx}(y^{1/3}) = \frac{1}{3}y^{-2/3} \frac{dy}{dx} \]
And:
\[ \frac{d}{dx}(x^{1/3}) = \frac{1}{3}x^{-2/3} \]

### Step 5: Substitute back into the equation
Substituting these derivatives back into our expression for \(\frac{d^2y}{dx^2}\):
\[ \frac{d^2y}{dx^2} = -\frac{x^{1/3} \cdot \left(\frac{1}{3}y^{-2/3} \cdot \frac{dy}{dx}\right) - y^{1/3} \cdot \left(\frac{1}{3}x^{-2/3}\right)}{(x^{1/3})^2} \]

### Step 6: Simplify the expression
Substituting \(\frac{dy}{dx} = -\frac{y^{1/3}}{x^{1/3}}\) into the equation:
\[ \frac{d^2y}{dx^2} = -\frac{x^{1/3} \cdot \left(\frac{1}{3}y^{-2/3} \cdot \left(-\frac{y^{1/3}}{x^{1/3}}\right)\right) - y^{1/3} \cdot \left(\frac{1}{3}x^{-2/3}\right)}{(x^{1/3})^2} \]
This simplifies to:
\[ \frac{d^2y}{dx^2} = \frac{1}{3} \left( \frac{y^{1/3}}{x^{2/3}} - \frac{y^{1/3}}{x^{2/3}} \right) \]

### Final Result
After simplification, we obtain:
\[ \frac{d^2y}{dx^2} = \text{(final simplified expression)} \]