The derivative of which function is 1/(1+x^2)?

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Multiple Choice

The derivative of which function is 1/(1+x^2)?

Explanation:
Derivatives of inverse trig functions: if y = arctan x, then tan y = x. Differentiating gives sec^2 y · dy/dx = 1, so dy/dx = 1/sec^2 y = cos^2 y. Using tan y = x, we have cos^2 y = 1/(1 + tan^2 y) = 1/(1 + x^2). Therefore dy/dx = 1/(1 + x^2). This matches the derivative of arctan x. The other options don’t fit: ln(1+x^2) differentiates to 2x/(1+x^2), x^2 to 2x, and 1/(1+x^2) to -2x/(1+x^2)^2, none of which equal 1/(1+x^2).

Derivatives of inverse trig functions: if y = arctan x, then tan y = x. Differentiating gives sec^2 y · dy/dx = 1, so dy/dx = 1/sec^2 y = cos^2 y. Using tan y = x, we have cos^2 y = 1/(1 + tan^2 y) = 1/(1 + x^2). Therefore dy/dx = 1/(1 + x^2).

This matches the derivative of arctan x. The other options don’t fit: ln(1+x^2) differentiates to 2x/(1+x^2), x^2 to 2x, and 1/(1+x^2) to -2x/(1+x^2)^2, none of which equal 1/(1+x^2).

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