Learn how to find the derivative of ln(sqrt x). The derivative is d/dx[ln(sqrt x)].
Below is the graph of f(x) = ln(sqrt 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.
ln(sqrt x) is a logarithmic function involving a square root. The square root can be rewritten as x^(1/2), which makes it clear that the Logarithmic Rule should be applied.
💡 Why this works: The presence of a logarithmic function and a square root indicates that we need to apply the Logarithmic Rule. This rule applies when differentiating the natural logarithm of any expression, and here it helps us differentiate ln(sqrt x).
To differentiate ln(sqrt x), we apply the Logarithmic Rule. The first step is to recognize that ln(sqrt x) is equivalent to ln(x^(1/2)). Then, we differentiate using the chain rule.
💡 Why this works: By applying the Logarithmic Rule, we get the derivative of ln(x^(1/2)) as (1/(x^(1/2))) * (d/dx[x^(1/2)]). The chain rule is used to differentiate the inner function x^(1/2).
Simplifying the result gives us the derivative of ln(sqrt x). After applying the chain rule, the result is (1/(2x^(1/2))).
💡 Why this works: The final derivative d/dx[ln(sqrt x)] is (1/(2sqrt x)), confirming that the differentiation steps and rules were applied correctly. This form is the simplest and most accurate representation of the derivative.
[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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