Find `dy/dx if x+3y-5=0`

Answer

To find \(\frac{dy}{dx}\) for the equation \(x + 3y - 5 = 0\), we will follow these steps:

### Step 1: Start with the given equation
We have the equation:
\[ x + 3y - 5 = 0 \]

### Step 2: Differentiate both sides with respect to \(x\)
We will differentiate each term in the equation with respect to \(x\):
- The derivative of \(x\) with respect to \(x\) is \(1\).
- The derivative of \(3y\) with respect to \(x\) requires the use of the chain rule. Since \(y\) is a function of \(x\), we have:
\[ \frac{d}{dx}(3y) = 3 \frac{dy}{dx} \]
- The derivative of the constant \(-5\) is \(0\).

Putting this all together, we differentiate the equation:
\[ \frac{d}{dx}(x) + \frac{d}{dx}(3y) - \frac{d}{dx}(5) = 0 \]
This gives us:
\[ 1 + 3\frac{dy}{dx} - 0 = 0 \]

### Step 3: Solve for \(\frac{dy}{dx}\)
Now, we can rearrange the equation to solve for \(\frac{dy}{dx}\):
\[ 1 + 3\frac{dy}{dx} = 0 \]
Subtract \(1\) from both sides:
\[ 3\frac{dy}{dx} = -1 \]
Now, divide both sides by \(3\):
\[ \frac{dy}{dx} = -\frac{1}{3} \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -\frac{1}{3} \]

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