Learn how to find the derivative of tan(x^2). The derivative is d/dx[tan(x^2)].
Below is the graph of f(x) = tan(x^2) 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.
In tan(x^2), we recognize that the argument of the tangent function is not just x, but x squared. This requires applying the Power Rule to the argument (x^2) first, and then differentiating the tangent function.
💡 Why this works: We use the Power Rule because the function tan(x^2) is composed of a composite function where the inside function is x^2. The Power Rule helps simplify the derivative process.
To differentiate tan(x^2), first apply the chain rule, as the argument of the tangent function is x^2. Then differentiate the tangent function itself and multiply by the derivative of x^2.
💡 Why this works: Differentiating tan(u) results in sec^2(u), where u is the inside function x^2. Multiply this by the derivative of x^2, which is 2x, giving us the derivative d/dx[tan(x^2)] = 2x * sec^2(x^2).
Now that we have the derivative expression, we simplify it to the final form: 2x * sec^2(x^2). This shows how the rate of change of tan(x^2) depends on both x and the secant squared of x^2.
💡 Why this works: This final expression is verified by ensuring all differentiation rules were applied correctly. It is the correct form for the derivative of tan(x^2).
[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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