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

Answer

To find \(\frac{dy}{dx}\) for the function \(y = \frac{\tan x}{x}\), we will apply 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 \(\frac{dy}{dx}\) is given by:

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

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

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

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

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

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

Substituting the values we have:

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

### Step 4: Simplify the Expression
This simplifies to:

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

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

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