The derivative d/dx arctan x equals which expression?

The derivative d/dx arctan x equals which expression?

• A. arctan x
• B. x/(1+x^2)
• C. 1/(1-x^2)
• D. 1/(1+x^2)

Answer: (D)

The key idea is using the derivative of an inverse trig function. Let y = arctan x, so tan y = x. Differentiate both sides with respect to x: sec^2 y · dy/dx = 1. Using sec^2 y = 1 + tan^2 y and tan y = x, we get dy/dx = 1/(1 + x^2). Therefore, the derivative is 1/(1 + x^2). This matches the fact that arctan x is the inverse of tan on (-pi/2, pi/2). The other expressions don’t fit this relationship: the derivative isn’t arctan x, nor x/(1+x^2), and 1/(1−x^2) corresponds to a different inverse function.

The key idea is using the derivative of an inverse trig function. Let y = arctan x, so tan y = x. Differentiate both sides with respect to x: sec^2 y · dy/dx = 1. Using sec^2 y = 1 + tan^2 y and tan y = x, we get dy/dx = 1/(1 + x^2). Therefore, the derivative is 1/(1 + x^2). This matches the fact that arctan x is the inverse of tan on (-pi/2, pi/2). The other expressions don’t fit this relationship: the derivative isn’t arctan x, nor x/(1+x^2), and 1/(1−x^2) corresponds to a different inverse function.