Kerala KEAM 5 June 2024 Solved Paper

We are given that the function $f(x)$ is defined as:$$f(x) = \frac{ | x | }{1 + | x | }$$Step 1: Differentiating the functionThe derivative of $f(x)$ depends on whether $x$ is positive or negative, since the absolute value function changes its expression depending on the sign of $x$.- For $x > 0$, $| x | = x$, so:$$f(x) = \frac{x}{1 + x}$$The derivative of this is:$$f'(x) = \frac{(1+x) \cdot 1 - x \cdot 1}{(1+x)^2} = \frac{1}{(1+x)^2}$$- For $x < 0$, $| x | = -x$, so:$$f(x) = \frac{-x}{1 - x}$$The derivative of this is:$$f'(x) = \frac{(1-x) \cdot (-1) - (-x) \cdot (-1)}{(1-x)^2} = \frac{-2x}{(1-x)^2}$$Step 2: Finding $f'(-2)$Since we are asked to find $f'(-2)$, we use the derivative for $x < 0$:$$f'(-2) = \frac{-2(-2)}{(1 - (-2))^2} = \frac{4}{(1 + 2)^2} = \frac{4}{9}$$Thus, the correct answer is $\frac{-4}{9}$. Therefore, the correct answer is option (C). Quick Tip: When differentiating piecewise functions involving absolute values, break the function into two cases: one for positive values of $x$ and one for negative values. Then apply the appropriate derivative formula for each case.