Learn how to find the derivative of x ln x. The derivative is d/dx[x ln x].
Below is the graph of f(x) = x ln x and its derivative f'(x). Notice how the original function and derivative relate to each other.
Original Function:
Derivative:
This derivative represents the rate of change of the original function. The calculation follows standard differentiation rules as shown in the step-by-step solution below.
To differentiate x ln x, we first identify the structure of the function. It’s a product of two functions: x and ln x. The Power Rule is applied here as it handles terms where x is raised to a power, combined with the logarithmic function.
💡 Why this works: The structure of x ln x calls for applying the product rule, not just the Power Rule. The Power Rule is used in the differentiation of x, which is considered a term of the form x^1.
Using the product rule, we differentiate both parts: x and ln x. The derivative of x is 1, and the derivative of ln x is 1/x. Applying the product rule, we get the derivative of x ln x as 1 * ln x + x * (1/x), which simplifies to ln x + 1.
💡 Why this works: The differentiation process leads to the simplified form of the derivative, which combines both the linear and logarithmic terms. This is why we see the final result as ln x + 1.
After applying the differentiation rules, the result is ln x + 1. This final form represents the rate of change of the function x ln x with respect to x, showing a simple but effective result.
💡 Why this works: We verify that d/dx[x ln x] = ln x + 1 is correct by ensuring the product rule and Power Rule were applied accurately. This confirms the correctness of the derivative.
[Teaching explanation - to be filled]
[Application to this function - to be filled]
❌ Common Mistake:
Not simplifying the final answer
✅ Correct Approach:
Always simplify your derivative to its most reduced form
Double-check your work by comparing your steps with our solution. Use our calculator to verify each step of the differentiation process.
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