If y = tan^(-1)(x), then what is the second derivative d^2y/dx^2?

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Question: If y = tan^(-1)(x), then what is the second derivative d^2y/dx^2?

Options:

Correct Answer: -2/(1+x^2)^2

Solution:

The first derivative dy/dx = 1/(1+x^2). The second derivative d^2y/dx^2 = -2/(1+x^2)^2.

If y = tan^(-1)(x), then what is the second derivative d^2y/dx^2?

If y = tan^(-1)(x), then what is the second derivative d^2y/dx^2?

Step 1: Start with the equation y = tan^(-1)(x). This means y is the inverse tangent of x.
Step 2: To find the first derivative dy/dx, use the derivative formula for inverse tangent: dy/dx = 1/(1+x^2).
Step 3: Now, we need to find the second derivative d^2y/dx^2. This means we need to differentiate dy/dx again.
Step 4: Differentiate dy/dx = 1/(1+x^2) using the quotient rule or chain rule.
Step 5: The derivative of 1/(1+x^2) is -2x/(1+x^2)^2.
Step 6: Since we want d^2y/dx^2, we need to express it in terms of x. We already have -2x/(1+x^2)^2.
Step 7: The final answer for the second derivative is d^2y/dx^2 = -2/(1+x^2)^2.

Inverse Trigonometric Functions – Understanding the properties and derivatives of inverse trigonometric functions, specifically the derivative of arctan.
Chain Rule – Applying the chain rule for differentiation when dealing with composite functions.
Higher Order Derivatives – Calculating second derivatives and understanding their significance in the context of the function's behavior.