Derivative of x*ln(x)

Step-by-step solution

1. 1Product Rule
   $$\frac{d}{dx}\left[x\cdot \ln \left(x\right)\right]=\frac{d}{dx}\left[x\right]\cdot \ln \left(x\right)+x\cdot \frac{d}{dx}\left[\ln \left(x\right)\right]$$

   Let $f=x$ and $g=\ln \left(x\right)$. The product rule says $(f\cdot g{)}^{\prime }=f^{\prime }\cdot g+f\cdot g^{\prime }$ — find each factor's derivative, then assemble.

2. ↳Find f′: Differentiate the first factor, $f=x$:
3. 2Variable Rule
   $$\frac{d}{dx}\left[x\right]=1$$

   Differentiating $x$ with respect to itself always gives $1$.

4. ↳Find g′: Differentiate the second factor, $g=\ln \left(x\right)$:
5. 3Derivative of ln
   $$\frac{d}{dx}\left[\ln \left(x\right)\right]=\frac{1}{x}$$

   The derivative of $\ln (x)$ is $\frac{1}{x}$ — a fundamental logarithm rule.

6. =Combine
   $$\frac{d}{dx}\left[x\cdot \ln \left(x\right)\right]=1\cdot \ln \left(x\right)+x\cdot \frac{1}{x}=\ln \left(x\right)+1$$

   With $f^{\prime }=1$ and $g^{\prime }=\frac{1}{x}$, substitute into $f^{\prime }\cdot g+f\cdot g^{\prime }$ and simplify.

7. ✓Final Answer
   $$\frac{d}{dx}\left[x\cdot \ln \left(x\right)\right]=\ln \left(x\right)+1$$

   The first derivative simplifies to $\ln \left(x\right)+1$.