Learn how to find the derivative of x/y. The derivative is d/dx[x/y].
Below is the graph of f(x) = x/y 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 f(x) = x/y is a quotient of two functions: x (the numerator) and y (the denominator). The Quotient Rule applies here because we are differentiating a function that is the ratio of two other functions.
💡 Why this works: The Quotient Rule is designed for precisely this kind of situation where one function is divided by another, and both need to be differentiated separately.
To differentiate f(x) = x/y using the Quotient Rule, we follow the formula: d/dx[f(x)] = (v * d/dx[u] - u * d/dx[v]) / v^2, where u = x and v = y.
💡 Why this works: Applying the Quotient Rule, the derivative of x/y is computed as (y * d/dx[x] - x * d/dx[y]) / y^2. This ensures both the numerator and denominator are differentiated correctly.
After applying the Quotient Rule, we simplify the resulting expression for the derivative. We verify the correctness by checking that both the numerator and denominator are handled according to the differentiation rules.
💡 Why this works: The final derivative expression, d/dx[x/y] = (y - x * d/dx[y]) / y^2, accurately reflects the rate of change of the function x/y. It is correct as it follows the Quotient Rule's structure.
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❌ 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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