Learn how to find the derivative of ln(sin x). The derivative is d/dx[ln(sin x)].
Below is the graph of f(x) = ln(sin 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.
To differentiate ln(sin x), we need to recognize the function as a logarithmic function applied to a trigonometric function. The Logarithmic Rule is appropriate here, as it allows us to differentiate the logarithm of a function.
💡 Why this works: Since the function is in the form ln(g(x)), where g(x) = sin(x), the Logarithmic Rule applies. This tells us we will need to differentiate the outer logarithmic function and then handle the inner function, sin(x), separately.
First, we apply the Logarithmic Rule, which states that the derivative of ln(f(x)) is f'(x)/f(x). In this case, the inner function is sin(x), so we differentiate it and divide by sin(x).
💡 Why this works: The derivative of ln(sin x) is the derivative of sin(x), which is cos(x), divided by sin(x). This gives us the expression cos(x)/sin(x), which simplifies to cot(x). Thus, the derivative of ln(sin x) is cot(x).
After applying the rules, we obtain the derivative cos(x)/sin(x), which simplifies to cot(x). This is the final derivative.
💡 Why this works: The final form of the derivative, cot(x), is the simplest expression representing the rate of change of ln(sin x). It’s essential to verify that each step in the differentiation process is correct to confirm this result.
[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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