Find `dy/dx if y= xtanx`

Answer

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

\[ \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \]

In our case, we can identify:
- \(u = x\)
- \(v = \tan x\)

Now, let's differentiate step by step.

### Step 1: Differentiate \(u\) and \(v\)
1. Differentiate \(u = x\):
\[ \frac{du}{dx} = 1 \]

2. Differentiate \(v = \tan x\):
\[ \frac{dv}{dx} = \sec^2 x \]

### Step 2: Apply the Product Rule
Now, we apply the product rule:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Substituting the values we found:
\[ \frac{dy}{dx} = x \cdot \sec^2 x + \tan x \cdot 1 \]

### Step 3: Simplify the Expression
Now we can simplify the expression:
\[ \frac{dy}{dx} = x \sec^2 x + \tan x \]

Thus, the final answer is:
\[ \frac{dy}{dx} = x \sec^2 x + \tan x \]