Find `(dy)/(dx)`, when:
`x^(n)+y^(n)=a^(n)`

Explanation

To find \(\frac{dy}{dx}\) for the equation \(x^n + y^n = a^n\), we will use implicit differentiation. Here’s the step-by-step solution:

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ x^n + y^n = a^n \]
Differentiating both sides with respect to \(x\):
\[ \frac{d}{dx}(x^n) + \frac{d}{dx}(y^n) = \frac{d}{dx}(a^n) \]

### Step 2: Apply the differentiation rules
Using the power rule for differentiation, we have:
\[ nx^{n-1} + n y^{n-1} \frac{dy}{dx} = 0 \]
Note that \(\frac{d}{dx}(a^n) = 0\) because \(a\) is a constant.

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

### Step 4: Solve for \(\frac{dy}{dx}\)
Now, we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = -\frac{nx^{n-1}}{ny^{n-1}} \]
The \(n\) in the numerator and denominator cancels out:
\[ \frac{dy}{dx} = -\frac{x^{n-1}}{y^{n-1}} \]

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

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