Learn how to find the derivative of 1/x^3. The derivative is d/dx[1/x^3].
Below is the graph of f(x) = 1/x^3 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^3 requires the Quotient Rule because it's a division of two functions. It can be rewritten as f(x) = 1/(x^3), which fits the structure for applying the Quotient Rule.
💡 Why this works: The Quotient Rule applies when a function is the quotient of two expressions. In this case, 1 is the numerator and x^3 is the denominator.
To differentiate 1/x^3, we apply the Quotient Rule and Power Rule. The Quotient Rule tells us to differentiate the numerator (1) and the denominator (x^3) and then combine the results.
💡 Why this works: By applying the Quotient Rule, the derivative becomes d/dx[1/x^3] = -3/x^4. The Power Rule simplifies x^3 to 3x^2, leading to the correct rate of change.
Simplify the result from Step 2 to get the derivative in its final form. The expression -3/x^4 is the simplified form of the derivative.
💡 Why this works: The negative sign comes from the differentiation of x^3, and the denominator is raised to the power of 4, as per the Power Rule. This result is the correct derivative for the function 1/x^3.
[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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