Learn how to find the derivative of tan(x^3). The derivative is d/dx[tan(x^3)].
Below is the graph of f(x) = tan(x^3) 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.
tan(x^3) is a composite function, where the outer function is tan(u) and the inner function is x^3. The Power Rule applies because the inner function, x^3, is a power of x.
💡 Why this works: In this case, the structure of tan(x^3) involves differentiating the outer tangent function while also considering the power rule for the inner function x^3.
The derivative of tan(u) is sec²(u). Using the chain rule, we differentiate the outer function (tan) first and then multiply by the derivative of the inner function (x^3).
💡 Why this works: The derivative of tan(x^3) involves first applying the derivative of tan(u), which is sec²(u), and then multiplying by the derivative of x^3, which is 3x².
The final derivative is the result of applying both the chain rule and the power rule. The expression becomes 3x² * sec²(x^3).
💡 Why this works: This simplification shows that the derivative of tan(x^3) is 3x² * sec²(x^3), which is the correct and final form. It combines the effects of both functions involved in the composition.
[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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