Question

The left hand derivative of $|x|$ with respect to $x$ at $x=0$ is

Hint:

For the modulus function \( |x| \), check the behavior separately on the left and right side of the point. At \(x=0\), the left hand derivative is \(-1\) and the right hand derivative is \(1\).

Answer 1

Step 1: Write the definition of the modulus function.
The absolute value function is defined as \[ |x|= \begin{cases} x, & x\ge 0 \\ -x, & x& Lt;0 \end{cases} \] Thus the function behaves differently on the left side and the right side of zero. 

Step 2: Understand the meaning of the left hand derivative. 
The left hand derivative at a point is the derivative evaluated as $x$ approaches the point from the left side. Mathematically it is written as \[ \lim_{h\to0^-}\frac{f(0+h)-f(0)}{h} \] This means we approach $0$ from negative values. 

Step 3: Use the definition of $|x|$ for $x<0$. 
For negative values of $x$ \[ |x|=-x \] Thus near the left side of zero the function behaves as \[ f(x)=-x \] 
Step 4: Differentiate the function. 
The derivative of \[ f(x)=-x \] is \[ f'(x)=-1 \] Thus the slope of the curve on the left side of zero is $-1$. 

Step 5: Conclusion. 
Therefore the left hand derivative of $|x|$ at $x=0$ is \[ -1 \] Final Answer: $\boxed{-1}$