Find `dy/dx if x-3y=x^4`

Answer

To find \(\frac{dy}{dx}\) for the equation \(x - 3y = x^4\), we can follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ x - 3y = x^4 \]

### Step 2: Isolate \(y\)
Rearranging the equation to express \(y\) in terms of \(x\):
\[ -3y = x^4 - x \]
\[ y = \frac{x - x^4}{3} \]

### Step 3: Differentiate both sides
Now, differentiate both sides with respect to \(x\):
\[ \frac{dy}{dx} = \frac{1}{3} \left( \frac{d}{dx}(x - x^4) \right) \]

### Step 4: Apply the differentiation rules
Using the power rule for differentiation:
\[ \frac{d}{dx}(x) = 1 \quad \text{and} \quad \frac{d}{dx}(x^4) = 4x^3 \]
Substituting these into the equation gives:
\[ \frac{dy}{dx} = \frac{1}{3} \left( 1 - 4x^3 \right) \]

### Step 5: Simplify the expression
Thus, we can simplify the expression for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{1 - 4x^3}{3} \]

### Final Answer
The final result is:
\[ \frac{dy}{dx} = \frac{1 - 4x^3}{3} \]

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