NDA 2026 — Mathematics PYQ

The given function is:

$$f(x)=\frac{x}{|x|}$$

1. Understanding the absolute value:

By definition, the absolute value function $|x|$ is defined as:

$$|x|=\{\begin{array}{llc}x& amp;\text{if }x\ge 0& \\ -x& amp;\text{if }x& lt;0\end{array}$$

2. Simplifying for the given condition (x < 0):

Since the problem specifically asks for the derivative when x < 0, we substitute $|x|=-x$ into the function:

$$f(x)=\frac{x}{-x}$$

$$f(x)=-1$$

3. Finding the derivative:

Now, we find the derivative of the constant function $f(x)=-1$ with respect to $x$:

$$\frac{d}{dx}(-1)=0$$

Conclusion:

The derivative of a constant value is always zero. Since the function simplifies to a constant value of $-1$ for all negative values of $x$, its derivative is $0$.

Final Answer:

The correct option is (b) 0.