Differentiate `x^(2)tanx`.

Answer

To differentiate the function \( y = x^2 \tan x \), 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 let:
- \( u = x^2 \)
- \( v = \tan x \)

Now, we need to find the derivatives of \( u \) and \( v \):

1. **Differentiate \( u = x^2 \)**:
\[ \frac{du}{dx} = 2x \]

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

Now, we can apply the product rule:

\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]

Substituting the values we have:

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

This simplifies to:

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

Thus, the derivative of \( y = x^2 \tan x \) is:

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