Find `dy/dx if y^5= x`

Answer

To find \(\frac{dy}{dx}\) given the equation \(y^5 = x\), we will use implicit differentiation. Here are the steps:

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ y^5 = x \]

2. **Differentiate both sides with respect to \(x\):**
- The left side requires the use of the chain rule since \(y\) is a function of \(x\):
\[ \frac{d}{dx}(y^5) = 5y^4 \frac{dy}{dx} \]
- The right side is straightforward:
\[ \frac{d}{dx}(x) = 1 \]

3. **Set the derivatives equal to each other:**
\[ 5y^4 \frac{dy}{dx} = 1 \]

4. **Solve for \(\frac{dy}{dx}\):**
\[ \frac{dy}{dx} = \frac{1}{5y^4} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{1}{5y^4} \]