Learn how to find the derivative of x/(x+1). The derivative is 1/(x+1)².
Below is the graph of f(x) = x/(x+1) 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 x/(x+1) is a ratio of two functions, where the numerator is x and the denominator is (x+1). This is the perfect setup for the Quotient Rule, which is used when differentiating a function that is a quotient of two other functions.
💡 Why this works: The Quotient Rule applies because the function is expressed as the ratio of two differentiable functions: x (the numerator) and (x+1) (the denominator). The rule tells us to differentiate the numerator and denominator separately and then combine them in a specific way.
To differentiate x/(x+1) using the Quotient Rule, we take the derivative of the numerator (which is 1) and multiply it by the denominator, (x+1), then subtract the product of the numerator, x, and the derivative of the denominator (which is also 1), divided by the square of the denominator.
💡 Why this works: Following the Quotient Rule, we get f'(x) = [(1)*(x+1) - (x)*(1)] / (x+1)², which simplifies to 1/(x+1)².
After applying the Quotient Rule, we simplify the expression to get the derivative f'(x) = 1/(x+1)². This form is the simplest and most accurate representation of the rate of change of the function.
💡 Why this works: The final result 1/(x+1)² is correct because the numerator simplifies to 1, and the denominator is the square of (x+1), as per the Quotient Rule. This is the simplest form of the derivative for this 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.
Calculate any derivative instantly with step-by-step solutions. Perfect for students, teachers, and anyone learning calculus.