Learn how to find the derivative of ln(cos x). The derivative is d/dx[ln(cos x)].
Below is the graph of f(x) = ln(cos 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.
The function ln(cos x) requires the Logarithmic Rule because it is a logarithm of a trigonometric function. In this case, we are differentiating the natural logarithm of cos x.
💡 Why this works: To differentiate this expression, the Logarithmic Rule must be applied first, followed by the chain rule to account for the derivative of cos x.
The Logarithmic Rule states that d/dx[ln(u)] = 1/u * du/dx. For ln(cos x), u = cos x, so we need to differentiate cos x.
💡 Why this works: By applying the rule, we get d/dx[ln(cos x)] = 1/cos x * (-sin x). This results in -tan x, which is the derivative of ln(cos x).
After applying the Logarithmic Rule and the chain rule, we simplify the result to -tan x, the derivative of ln(cos x).
💡 Why this works: This simplification follows directly from the application of differentiation rules and is verified as correct by checking with other methods of differentiation.
[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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