Learn how to find the derivative of ln(tan x). The derivative is d/dx[ln(tan x)].
Below is the graph of f(x) = ln(tan 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(tan x) involves both a logarithmic function (ln) and a trigonometric function (tan x), which makes it a composite function. The natural logarithm of a trigonometric function requires the application of the Logarithmic Rule.
💡 Why this works: Since ln(tan x) is a composition of functions (ln and tan x), we use the Logarithmic Rule. The rule tells us that the derivative of ln(f(x)) is 1/f(x) times the derivative of f(x). This rule applies here because we are differentiating the natural logarithm of a function, tan x.
Now, we apply the Logarithmic Rule to ln(tan x). According to the rule, we differentiate the outer function (ln) first, treating tan x as the inside function.
💡 Why this works: Differentiating ln(u) where u = tan x gives us 1/tan x. Then, we multiply by the derivative of tan x, which is sec^2(x). Therefore, the derivative of ln(tan x) is 1/tan x * sec^2(x).
We simplify the expression to make it more interpretable. By using the identity sec^2(x) = 1 + tan^2(x), we can rewrite the derivative in a more manageable form.
💡 Why this works: The simplified derivative of ln(tan x) is sec^2(x)/tan x. This is the correct final form of the derivative, showing how the rate of change of ln(tan x) behaves with respect to x.
[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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