Learn how to find the derivative of ln(1/x). The derivative is d/dx[ln(1/x)].
Below is the graph of f(x) = ln(1/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.
The function ln(1/x) requires both the Quotient Rule and Logarithmic Rule for differentiation. The Quotient Rule applies due to the fraction 1/x, and the Logarithmic Rule is used to differentiate the natural logarithm.
💡 Why this works: ln(1/x) can be seen as the logarithm of a quotient, so the Quotient Rule helps to handle the fraction, and the Logarithmic Rule deals with the differentiation of the logarithmic part.
Apply the Logarithmic Rule to ln(1/x), which simplifies the expression to ln(x^-1). Then, the Quotient Rule differentiates the reciprocal of x. The derivative is obtained by combining both rules.
💡 Why this works: Differentiating ln(1/x) involves using the chain rule within the Quotient Rule. The derivative of ln(x^-1) is -1/x, so the final derivative of ln(1/x) is -1/x.
After applying the Quotient Rule and Logarithmic Rule, we simplify the expression to -1/x, which represents the rate of change of ln(1/x) with respect to x.
💡 Why this works: The simplified derivative, -1/x, is the final result. It verifies that the derivative follows the rules correctly and reflects the behavior of ln(1/x).
[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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