Find `(dy)/(dx)` when :
`y=(x+tanx)/(tanx)`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = \frac{x + \tan x}{\tan x}\), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form \(\frac{a}{b}\), then the derivative is given by:

\[ \frac{dy}{dx} = \frac{b \frac{da}{dx} - a \frac{db}{dx}}{b^2} \]

where \(a = x + \tan x\) and \(b = \tan x\).

### Step 1: Identify \(a\) and \(b\)
Let:
- \(a = x + \tan x\)
- \(b = \tan x\)

### Step 2: Differentiate \(a\) and \(b\)
Now we need to find \(\frac{da}{dx}\) and \(\frac{db}{dx}\).

1. Differentiate \(a\):
\[ \frac{da}{dx} = \frac{d}{dx}(x) + \frac{d}{dx}(\tan x) = 1 + \sec^2 x \]

2. Differentiate \(b\):
\[ \frac{db}{dx} = \frac{d}{dx}(\tan x) = \sec^2 x \]

### Step 3: Apply the Quotient Rule
Now, substitute \(a\), \(b\), \(\frac{da}{dx}\), and \(\frac{db}{dx}\) into the quotient rule formula:

\[ \frac{dy}{dx} = \frac{\tan x (1 + \sec^2 x) - (x + \tan x)(\sec^2 x)}{\tan^2 x} \]

### Step 4: Simplify the Expression
Now, simplify the numerator:

1. Expand the numerator:
\[ \tan x + \tan x \sec^2 x - x \sec^2 x - \tan x \sec^2 x \]

2. Combine like terms:
\[ \tan x - x \sec^2 x \]

Thus, the expression for \(\frac{dy}{dx}\) becomes:
\[ \frac{dy}{dx} = \frac{\tan x - x \sec^2 x}{\tan^2 x} \]

### Final Answer
The derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{\tan x - x \sec^2 x}{\tan^2 x} \]