Find the derivative of the function: $y=arctan\frac{1-x}{1+x}$.

• A. $\frac{-1}{1+x^2}$
• B. $\frac{2}{{(1+x)}^2+{(1-x)}^2}$
• C. $\frac{1}{1+x^2}$
• D. $\frac{-2}{{(1+x)}^2+{(1-x)}^2}$

Answer: (D)

Verified step by step guidance

1. Step 1: Recognize that the function y = arctan(\(\frac{1 - x}{1 + x}\)) involves the arctan function, which has a derivative formula. Recall that the derivative of arctan(u) with respect to x is \(\frac{du/dx}{1 + u^2}\).
2. Step 2: Identify u = \(\frac{1 - x}{1 + x}\) as the argument of the arctan function. To apply the derivative formula, first compute the derivative of u with respect to x. Use the quotient rule: \(\frac{d}{dx}\[\left\)(\(\frac{1 - x}{1 + x}\]\right\)) = \(\frac{(1 + x)(-1) - (1 - x)(1)}{(1 + x)^2}\).
3. Step 3: Simplify the numerator of the derivative of u. Combine terms: (-1)(1 + x) - (1)(1 - x) = -1 - x - 1 + x = -2. Thus, \(\frac{du}{dx}\) = \(\frac{-2}{(1 + x)^2}\).
4. Step 4: Substitute \(\frac{du}{dx}\) and u = \(\frac{1 - x}{1 + x}\) into the derivative formula for arctan(u). The derivative becomes \(\frac{-2}{(1 + x)^2}\) / \(\left\)(1 + \(\left\)(\(\frac{1 - x}{1 + x}\)\(\right\))^2\(\right\)).
5. Step 5: Simplify the denominator 1 + \(\left\)(\(\frac{1 - x}{1 + x}\)\(\right\))^2. Square \(\frac{1 - x}{1 + x}\) to get \(\frac{(1 - x)^2}{(1 + x)^2}\), then add 1: \(\frac{(1 + x)^2 + (1 - x)^2}{(1 + x)^2}\). Combine this with the numerator \(\frac{-2}{(1 + x)^2}\) to get the final derivative expression: \(\frac{-2}{(1 + x)^2 + (1 - x)^2}\).