derivative of xlnx
Understand the Problem
The question is asking for the derivative of the function f(x) = x ln(x) with respect to x. To solve this, we will use the product rule of differentiation, since it involves two functions: x and ln(x).
Answer
The derivative of $f(x) = x \ln(x)$ is $f'(x) = \ln(x) + 1$.
Answer for screen readers
The derivative of the function $f(x) = x \ln(x)$ with respect to $x$ is: $$ f'(x) = \ln(x) + 1 $$
Steps to Solve
- Identify the functions for the product rule
The function $f(x) = x \ln(x)$ has two parts: Let $u = x$ and $v = \ln(x)$.
- Apply the product rule
The product rule states that if you have two functions $u$ and $v$, the derivative of their product is given by: $$ f'(x) = u'v + uv' $$
- Calculate the derivatives of each function
Now we need to find the derivatives of $u$ and $v$:
- For $u = x$, its derivative $u' = 1$.
- For $v = \ln(x)$, its derivative $v' = \frac{1}{x}$.
- Substitute the derivatives into the product rule formula
Now, plug the derivatives into the product rule: $$ f'(x) = (1)(\ln(x)) + (x)\left(\frac{1}{x}\right) $$
- Simplify the expression
Now we simplify the expression: $$ f'(x) = \ln(x) + 1 $$
The derivative of the function $f(x) = x \ln(x)$ with respect to $x$ is: $$ f'(x) = \ln(x) + 1 $$
More Information
The derivative of the function gives us the slope of the tangent line at any point on the function $f(x) = x \ln(x)$. This is useful in various applications, including optimization problems in calculus.
Tips
- Failing to apply the product rule correctly by not identifying both functions properly.
- Forgetting to differentiate both components of the product.
- Not simplifying the final answer correctly.
