Learn how to find the derivative of 1/(x+1). The derivative is d/dx[1/(x+1)].
Below is the graph of f(x) = 1/(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 1/(x+1) consists of a constant 1 divided by a linear expression (x+1), making it a rational function. This structure directly points to the use of the Quotient Rule, which is designed for differentiating the ratio of two functions.
💡 Why this works: To apply the Quotient Rule, we identify the numerator (1) and the denominator (x+1). The Quotient Rule applies because we are differentiating a quotient of two functions.
The Quotient Rule states that the derivative of a quotient is given by (d/dx[u/v] = (v*u' - u*v') / v^2). Here, u = 1 and v = x + 1. The derivative of u is 0 and the derivative of v is 1.
💡 Why this works: Applying the Quotient Rule, we get: d/dx[1/(x+1)] = [(x+1)(0) - (1)(1)] / (x+1)^2, which simplifies to -1 / (x+1)^2.
After applying the Quotient Rule, we simplify the result: -1 / (x+1)^2. This is the derivative of 1/(x+1), which gives the rate of change of the function.
💡 Why this works: This final form is correct because it accurately reflects the differentiation rules applied to the function 1/(x+1), confirming that the derivative is -1 / (x+1)^2.
[Teaching explanation - to be filled]
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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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