find f'(x), f''(x) and f'''(x) for x\ln\left(x\right)

We are asked to find the first, second, and third derivatives of the function \( f(x) = x \ln(x) \).

### Step 1: First Derivative \( f'(x) \)
We use the **product rule** to differentiate \( f(x) = x \ln(x) \).

The product rule states that:
\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]
Here, \( u(x) = x \) and \( v(x) = \ln(x) \).

Now, calculate their derivatives:
- \( u'(x) = \frac{d}{dx}[x] = 1 \)
- \( v'(x) = \frac{d}{dx}[\ln(x)] = \frac{1}{x} \)

Now, apply the product rule:
\[
f'(x) = 1 \cdot \ln(x) + x \cdot \frac{1}{x} = \ln(x) + 1
\]

### Step 2: Second Derivative \( f''(x) \)
Now we differentiate \( f'(x) = \ln(x) + 1 \).

The derivative of \( \ln(x) \) is \( \frac{1}{x} \), and the derivative of 1 is 0. So:
\[
f''(x) = \frac{1}{x}
\]

### Step 3: Third Derivative \( f'''(x) \)
Finally, we differentiate \( f''(x) = \frac{1}{x} \).

The derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \). So:
\[
f'''(x) = -\frac{1}{x^2}
\]

### Summary:
\[
f'(x) = \ln(x) + 1
\]
\[
f''(x) = \frac{1}{x}
\]
\[
f'''(x) = -\frac{1}{x^2}
\]

Would you like further details on these steps or have any questions?

Here are 5 related questions you might consider:
1. How does the product rule apply to more complex functions?
2. What are the conditions for \( f'(x) \) to exist for logarithmic functions?
3. Can the chain rule be used in differentiating \( x \ln(x) \)?
4. What happens to higher-order derivatives of \( x \ln(x) \)?
5. How do the properties of logarithms affect their derivatives?

**Tip:** When differentiating a product of two functions, always consider applying the product rule first to avoid mistakes!

find f'(x), f''(x) and f'''(x) for x\ln(x)

Learn how to find the first, second, and third derivatives of the function f(x) = xln(x) using the product rule and logarithmic differentiation. This step-by-step guide explains key calculus concepts for higher-order derivatives, suitable for advanced high school and college students.

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