To find the derivative of the function \( y = x \ln x \), we can use the product rule of differentiation. The product rule states that if you have a function that is the product of two functions, say \( u \) and \( v \), then the derivative \( \frac{dy}{dx} \) is given by:
\[
\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}
\]
In our case, we can assign:
- \( u = x \)
- \( v = \ln x \)
Now, we will differentiate each part:
1. **Differentiate \( u \)**:
\[
\frac{du}{dx} = \frac{d}{dx}(x) = 1
\]
2. **Differentiate \( v \)**:
\[
\frac{dv}{dx} = \frac{d}{dx}(\ln x) = \frac{1}{x}
\]
Now, we can apply the product rule:
\[
\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}
\]
Substituting the values we have:
\[
\frac{dy}{dx} = x \cdot \frac{1}{x} + \ln x \cdot 1
\]
Now, simplifying this expression:
\[
\frac{dy}{dx} = 1 + \ln x
\]
Thus, the derivative \( \frac{dy}{dx} \) is:
\[
\frac{dy}{dx} = 1 + \ln x
\]
### Final Answer:
\[
\frac{dy}{dx} = 1 + \ln x
\]
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To find the derivative of the function \( y = x \ln x \), we can use the product rule of differentiation. The product rule states that if you have a function that is the product of two functions, say \( u \) and \( v \), then the derivative \( \frac{dy}{dx} \) is given by:
\[
\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}
\]
In our case, we can assign:
- \( u = x \)
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