Differentiability

Step 1: Express in terms of
. By Componendo and Dividendo: . .
Step 2: Differentiate
. . .
Final Answer: (b)

Step 1: Substitution
Let . Then and .
Step 2: Simplify Functions
. .
Step 3: Derivative
Since , then .
Final Answer: (c)

Step 1: Integrate RHS
Integrate . . .
Step 2: Match Coefficients
Comparing with : and .
Step 3: Calculate Result
.
Final Answer: (b)

Step 1: Recall the definition of differentiability.
A function is differentiable at a point if its derivative exists at that point. The absolute value function is piecewise-defined:

Step 2: Differentiate the function for and .
For , , so the derivative is:
For , , so the derivative is:


Step 3: Analyze the behavior at .
At , we must check the left-hand and right-hand derivatives. The left-hand derivative (approaching 0 from the negative side) is:
The right-hand derivative (approaching 0 from the positive side) is:

Step 4: Conclusion.
Since the left-hand and right-hand derivatives at are not equal, the function is not differentiable at . However, the function is differentiable everywhere else. Therefore, the set of points where is differentiable is , corresponding to option (A).

Step 5: Verification.
The absolute value function is continuous everywhere, but it is only non-differentiable at because of the discontinuity in the derivative.

Step 1: Recognize the function and its components.
We are given the function . The absolute value function implies that the function behaves differently for positive and negative values of . We must differentiate the function piecewise.

Step 2: Differentiate the function for .
When , we have: Using the product rule, the derivative is:

Step 3: Differentiate the function for .
When , we have: Again, using the product rule, the derivative is:

Step 4: Apply the absolute value.
The function is piecewise defined, and we need to adjust the derivative for each case. Therefore, for both and , the derivative is the same:

Step 5: Conclusion.
Thus, the derivative of is , corresponding to option (A).

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}
LHS:
RHS:


Step 3: Final Answer

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}



(circles) or (rectangular hyperbolas)
Step 3: Final Answer