How to differentiate x ln x?
Understand the Problem
The question is asking how to find the derivative of the function x ln(x) with respect to x. To solve this, we can use the product rule of differentiation, since it is the product of two functions: x and ln(x).
Answer
The derivative is $ \ln(x) + 1 $.
Answer for screen readers
The derivative of the function $x \ln(x)$ with respect to $x$ is: $$ \frac{d}{dx}(x \ln(x)) = \ln(x) + 1 $$
Steps to Solve
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Identify the functions We have two functions in the product: $u = x$ and $v = \ln(x)$.
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Apply the Product Rule The product rule states that if you have two functions $u$ and $v$, the derivative is given by: $$ \frac{d}{dx}(uv) = u'v + uv'$$
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Differentiate the functions Now we need to find the derivatives of $u$ and $v$.
- The derivative of $u = x$ is $u' = 1$.
- The derivative of $v = \ln(x)$ is $v' = \frac{1}{x}$.
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Substitute into the Product Rule Now plug $u$, $u'$, $v$, and $v'$ into the product rule: $$ \frac{d}{dx}(x \ln(x)) = (1) \cdot \ln(x) + (x) \cdot \left(\frac{1}{x}\right) $$
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Simplify the expression Now simplify the expression we derived: $$ \frac{d}{dx}(x \ln(x)) = \ln(x) + 1 $$
The derivative of the function $x \ln(x)$ with respect to $x$ is: $$ \frac{d}{dx}(x \ln(x)) = \ln(x) + 1 $$
More Information
The derivative represents the rate of change of the function $x \ln(x)$ with respect to $x$. This is important in calculus and has applications in optimization and analysis of functions. The product rule is a key tool in finding derivatives for products of functions.
Tips
- Forgetting to apply the product rule correctly by not including both parts of the product in the final derivative expression.
- Mixing up the derivative of $\ln(x)$, which is $\frac{1}{x}$, and incorrectly using it.
