To find the derivative of the function \( y = x \ln x \), 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:
\[
\frac{d(uv)}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}
\]
In our case, we can identify:
- \( u = x \)
- \( v = \ln x \)
Now, we will differentiate each function:
1. The derivative of \( u \) with respect to \( x \):
\[
\frac{du}{dx} = 1
\]
2. The derivative of \( v \) with respect to \( x \):
\[
\frac{dv}{dx} = \frac{1}{x}
\]
Now, applying the product rule:
\[
\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}
\]
Substituting the values of \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \):
\[
\frac{dy}{dx} = x \cdot \frac{1}{x} + \ln x \cdot 1
\]
Now simplify this expression:
\[
\frac{dy}{dx} = 1 + \ln x
\]
Thus, the derivative \( \frac{dy}{dx} \) is:
\[
\frac{dy}{dx} = 1 + \ln x
\]
### Final Answer:
\[
\frac{dy}{dx} = 1 + \ln x
\]
To find the derivative of the function \( y = x \ln x \), 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:
\[
\frac{d(uv)}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}
\]
In our case, we can identify:
- \( u = x \)
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