What is the derivative of xlnx - x?
Understand the Problem
The question is asking for the derivative of the function given by xlnx - x. To solve it, we will apply the rules of differentiation, including the product rule and the basic derivative of a constant and logarithmic functions.
Answer
The derivative is \( f'(x) = \ln x \).
Answer for screen readers
The derivative of the function ( f(x) = x \ln x - x ) is ( f'(x) = \ln x ).
Steps to Solve
- Identify the function to differentiate
We have the function ( f(x) = x \ln x - x ).
- Differentiate each term separately
To find the derivative ( f'(x) ), we differentiate the terms ( x \ln x ) and ( -x ) separately.
- Apply the product rule to ( x \ln x )
For the term ( x \ln x ), we need to use the product rule, which states that if ( u = x ) and ( v = \ln x ), then
$$ \frac{d}{dx}(uv) = u'v + uv' $$
where ( u' = 1 ) and ( v' = \frac{1}{x} ).
Applying the product rule gives us:
$$ \frac{d}{dx}(x \ln x) = 1 \cdot \ln x + x \cdot \frac{1}{x} = \ln x + 1 $$
- Differentiate the constant term ( -x )
The derivative of ( -x ) is simply
$$ \frac{d}{dx}(-x) = -1 $$
- Combine the derivatives
So now we combine both results: $$ f'(x) = \frac{d}{dx}(x \ln x) + \frac{d}{dx}(-x) = (\ln x + 1) - 1 $$
- Simplify the final expression
The final expression simplifies to:
$$ f'(x) = \ln x + 1 - 1 = \ln x $$
The derivative of the function ( f(x) = x \ln x - x ) is ( f'(x) = \ln x ).
More Information
The final derivative ( f'(x) = \ln x ) indicates how the function ( f(x) ) changes as ( x ) changes. The logarithmic derivative is particularly interesting in applications where rates of growth are analyzed.
Tips
- Forgetting to apply the product rule when differentiating the ( x \ln x ) term can lead to incorrect results.
- Not simplifying the expression after differentiating can leave the answer in a complicated form.
