Find `dy/dx if y= x/tanx`

Answer

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

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where \(u = x\) and \(v = \tan x\).

### Step 1: Identify \(u\) and \(v\)
Let:
- \(u = x\)
- \(v = \tan x\)

### Step 2: Differentiate \(u\) and \(v\)
Now we need to find the derivatives of \(u\) and \(v\):
- \(\frac{du}{dx} = 1\)
- \(\frac{dv}{dx} = \sec^2 x\) (since the derivative of \(\tan x\) is \(\sec^2 x\))

### Step 3: Apply the Quotient Rule
Now we can apply the quotient rule:

\[ \frac{dy}{dx} = \frac{\tan x \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{(\tan x)^2} \]

Substituting the values we found:

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

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

\[ \frac{dy}{dx} = \frac{\tan x - x \sec^2 x}{\tan^2 x} \]

### Final Result
Thus, the derivative \(\frac{dy}{dx}\) is:

\[ \frac{dy}{dx} = \frac{\tan x - x \sec^2 x}{\tan^2 x} \]