The derivative d/dx arctan x equals which expression?

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Multiple Choice

The derivative d/dx arctan x equals which expression?

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.

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