Learn how to find the derivative of 1/x. The derivative is d/dx[1/x].
Below is the graph of f(x) = 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 1/x is a ratio of two functions, where the numerator is 1 and the denominator is x. The Quotient Rule is necessary because it deals specifically with differentiating functions in the form of a quotient.
💡 Why this works: The Quotient Rule states that if you have a function in the form of f(x) = g(x)/h(x), its derivative is given by d/dx[g(x)/h(x)] = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. Since 1/x is a ratio, we apply this rule to find the derivative.
To differentiate 1/x using the Quotient Rule, set g(x) = 1 and h(x) = x. Differentiate both g(x) and h(x): g'(x) = 0 and h'(x) = 1. Plug these into the Quotient Rule formula.
💡 Why this works: Using the Quotient Rule, we have d/dx[1/x] = (0 * x - 1 * 1) / x^2. This simplifies to -1/x^2.
The result from Step 2 simplifies to -1/x^2. This is the derivative of 1/x. To verify, we can check that the application of the Quotient Rule was correct and that it matches the expected outcome for this type of function.
💡 Why this works: The derivative of 1/x is indeed -1/x^2, and this correctly represents the rate of change of the function 1/x with respect to x. This confirms that the Quotient Rule has been applied correctly.
[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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