Learn how to find the derivative of e^(ln x). The derivative is d/dx[e^(ln x)].
Below is the graph of f(x) = e^(ln 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 begin by recognizing that e^(ln x) can be simplified due to the properties of logarithms and exponents. Specifically, e^(ln x) simplifies to x, which is a polynomial function.
💡 Why this works: This structure suggests that we can apply the Power Rule, as x is in a form suitable for differentiation by this rule.
Using the Power Rule, the derivative of x is simply 1. This rule states that the derivative of x^n is n*x^(n-1), and since x is effectively x^1, we apply this rule directly.
💡 Why this works: This results in the derivative d/dx[e^(ln x)] = 1, demonstrating that the function’s rate of change is constant and does not vary with x.
After applying the Power Rule, the derivative is straightforward: d/dx[e^(ln x)] = 1. This is the correct result because the function e^(ln x) simplifies to x, and its rate of change is 1.
💡 Why this works: This verifies that the derivative is correctly computed, and the application of the Power Rule in this case is consistent with the expected outcome.
[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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