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

Answer

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

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ 3x - 5y = \sec x \]
Now, we differentiate both sides with respect to \(x\).

### Step 2: Apply differentiation rules
Using the rules of differentiation:
- The derivative of \(3x\) is \(3\).
- The derivative of \(-5y\) is \(-5 \frac{dy}{dx}\) (using the chain rule).
- The derivative of \(\sec x\) is \(\sec x \tan x\).

So, differentiating both sides gives us:
\[ \frac{d}{dx}(3x) - \frac{d}{dx}(5y) = \frac{d}{dx}(\sec x) \]
This results in:
\[ 3 - 5 \frac{dy}{dx} = \sec x \tan x \]

### Step 3: Solve for \(\frac{dy}{dx}\)
Now, we need to isolate \(\frac{dy}{dx}\). We can rearrange the equation:
\[ -5 \frac{dy}{dx} = \sec x \tan x - 3 \]
Dividing both sides by \(-5\) gives:
\[ \frac{dy}{dx} = \frac{3 - \sec x \tan x}{5} \]

### Final Result
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{3 - \sec x \tan x}{5} \]

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