Learn how to find the derivative of x^2 / x. The derivative is d/dx[x^2 / x].
Below is the graph of f(x) = x^2 / 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.
We need to recognize that x^2 / x is a rational function, which means it is the quotient of two functions. This requires the use of the Quotient Rule for differentiation. However, before applying the Quotient Rule, it is important to first simplify the expression.
💡 Why this works: The function x^2 / x simplifies to x, making the differentiation easier. But for clarity, we'll start by applying the Quotient Rule, which gives us the proper structure to find the derivative.
The Quotient Rule states that the derivative of a quotient u(x)/v(x) is given by (v(x)u'(x) - u(x)v'(x)) / (v(x))^2. For x^2 / x, we let u(x) = x^2 and v(x) = x. Differentiating these gives u'(x) = 2x and v'(x) = 1.
💡 Why this works: Using the Quotient Rule, the derivative becomes (x * 2x - x^2 * 1) / x^2. This simplifies further to (2x^2 - x^2) / x^2, which simplifies to x / x^2.
Now, simplify the result x / x^2. This simplifies to 1/x. Therefore, the derivative of x^2 / x is 1/x.
💡 Why this works: The derivative, 1/x, represents the rate of change of the function x^2 / x. The result confirms that the differentiation was performed correctly and matches the simplified form of the function.
[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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