Learn how to find the derivative of tan 3x. The derivative is d/dx[tan 3x].
Below is the graph of f(x) = tan 3x 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 tan 3x consists of a composite function where the outer function is tan(u) and the inner function is 3x. We recognize that the derivative involves applying the Power Rule in combination with the chain rule to account for the 3x inside the tangent.
💡 Why this works: The outer function, tan(u), differentiates to sec²(u), and the inner function 3x differentiates to 3. This composite structure means that the derivative will involve multiplying the derivative of tan with the derivative of 3x.
To differentiate tan 3x, we first differentiate the outer function, tan(u), to get sec²(3x). Next, we multiply by the derivative of the inner function, 3x, which is 3.
💡 Why this works: Thus, we apply the chain rule to get the derivative: d/dx[tan 3x] = 3 * sec²(3x). This is the result of differentiating tan(u) with u = 3x and applying the chain rule.
The final derivative is simplified to 3 * sec²(3x). There’s no further simplification needed since sec²(3x) is already in its simplest form.
💡 Why this works: This derivative gives us the rate of change of tan 3x at any point x. It’s the correct derivative because all differentiation rules were applied accurately, and it matches the structure of the 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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