Learn how to find the derivative of tan(ln x). The derivative is d/dx[tan(ln x)].
Below is the graph of f(x) = tan(ln 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 begin by recognizing that tan(ln x) involves both a trigonometric function (tan) and a logarithmic function (ln x). The composition of these functions requires the use of the Chain Rule in conjunction with the Power Rule for differentiation.
💡 Why this works: The Power Rule applies because we differentiate the outer function (tan) while considering the inner function (ln x) as a separate entity. This structure allows us to break down the derivative step by step.
To differentiate tan(ln x), we first apply the derivative of tan(u), which is sec²(u). Then, we differentiate the inner function, ln x, which gives 1/x. By the Chain Rule, we multiply these two results.
💡 Why this works: The derivative of tan(ln x) is therefore sec²(ln x) multiplied by the derivative of ln x, which is 1/x. This results in the full derivative: d/dx[tan(ln x)] = sec²(ln x) * (1/x).
After applying the differentiation rules, we have the expression d/dx[tan(ln x)] = sec²(ln x) * (1/x). There are no further simplifications possible, as this is the simplest form.
💡 Why this works: This is the correct final form of the derivative, and it can be verified by applying the Chain Rule and checking that the components of the derivative are correctly derived from the original function.
[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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