Learn how to find the derivative of arctan x. The derivative is d/dx[arctan x].
Below is the graph of f(x) = arctan 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.
The arctan x function is a type of inverse trigonometric function. To differentiate it, we apply the Power Rule, which is typically used for functions of the form x^n. Since arctan x is not directly in the form of a simple polynomial, we recognize that its derivative involves a formula derived from its inverse function structure.
💡 Why this works: The Power Rule works because arctan x behaves in a way similar to other functions that involve rational exponents. The inverse tangent function has a straightforward derivative structure based on these principles, leading to the formula for its rate of change.
To differentiate arctan x, we use the established derivative formula for inverse tangent. The Power Rule tells us that the derivative of arctan x is 1 / (1 + x^2).
💡 Why this works: The Power Rule is applied by recognizing that the change in arctan x as x changes is captured by 1 / (1 + x^2), reflecting how the slope of the function changes at every point along the curve.
After applying the derivative formula, we arrive at the simplified expression d/dx[arctan x] = 1 / (1 + x^2). This is the final derivative, which tells us how arctan x changes for any value of x.
💡 Why this works: This derivative is correct because it accurately reflects how the function behaves with respect to x. Its form, 1 / (1 + x^2), shows that the rate of change decreases as x becomes larger, which matches the known behavior of arctan 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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