If `x^(2//3)+y^(2//3)=a^(2//3)` , find `(dy)/(dx)`

Answer

To find \(\frac{dy}{dx}\) for the equation \(x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}\), we will differentiate both sides with respect to \(x\). Let's go through the steps:

### Step 1: Differentiate both sides of the equation
The given equation is:
\[ x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}} \]
Differentiating both sides with respect to \(x\):
\[ \frac{d}{dx}(x^{\frac{2}{3}}) + \frac{d}{dx}(y^{\frac{2}{3}}) = \frac{d}{dx}(a^{\frac{2}{3}}) \]

### Step 2: Apply the power rule
Using the power rule for differentiation, we get:
\[ \frac{2}{3}x^{-\frac{1}{3}} + \frac{2}{3}y^{-\frac{1}{3}} \frac{dy}{dx} = 0 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \(\frac{dy}{dx}\):
\[ \frac{2}{3}y^{-\frac{1}{3}} \frac{dy}{dx} = -\frac{2}{3}x^{-\frac{1}{3}} \]

### Step 4: Solve for \(\frac{dy}{dx}\)
Dividing both sides by \(\frac{2}{3}y^{-\frac{1}{3}}\):
\[ \frac{dy}{dx} = -\frac{x^{-\frac{1}{3}}}{y^{-\frac{1}{3}}} \]
This simplifies to:
\[ \frac{dy}{dx} = -\frac{y^{\frac{1}{3}}}{x^{\frac{1}{3}}} \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -\frac{y^{\frac{1}{3}}}{x^{\frac{1}{3}}} \]

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