Learn how to find the derivative of ln(1+sin x). The derivative is d/dx[ln(1+sin x)].
Below is the graph of f(x) = ln(1+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.
We recognize that ln(1+sin x) is a logarithmic function involving a composite function inside the logarithm. This requires the Logarithmic Rule, which applies when differentiating natural logarithms of functions. The presence of 1 + sin x inside the logarithm dictates the use of the chain rule to differentiate the inside expression.
💡 Why this works: The Logarithmic Rule requires us to differentiate the inside function separately. For ln(1+sin x), this means differentiating the expression 1 + sin x as a whole.
We differentiate ln(1+sin x) using the Logarithmic Rule, which states that d/dx[ln(u)] = 1/u * du/dx. In this case, u = 1 + sin x, and the derivative of u with respect to x is cos x.
💡 Why this works: Thus, applying the rule, the derivative of ln(1+sin x) becomes 1/(1+sin x) * cos x, which simplifies to d/dx[ln(1+sin x)] = cos x / (1 + sin x).
The result from Step 2, cos x / (1 + sin x), is the simplified form of the derivative. This is the final result for d/dx[ln(1+sin x)].
💡 Why this works: To verify, we check that the differentiation rules were applied correctly and that the expression matches the expected derivative form for logarithmic functions involving trigonometric components.
[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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