Differentiation

Step 1: Find the derivative of
The given function is . Differentiate using the quotient rule:
Step 2: Find critical points
For :
Step 3: Determine intervals of increase and decrease
For : For :
Step 4: Conclude the result
The function is strictly increasing on and strictly decreasing on .

Step 1: Find the limits at
For : For : At :
Step 2: Set up the continuity condition
For to be continuous at : Substitute the values: Step 3: Solve for and
From : From : Substitute into : Step 4: Conclude the result
The values of and are and .

Step 1: Rearrange the given equation
The given equation is: Divide throughout by : This implies:
Step 2: Differentiate with respect to
Differentiate both sides with respect to : Simplify:
Step 3: Express in terms of
Using , we get: Multiply through by :
Step 4: Conclude the result

Step 1: Rewrite the given equation
Given , take the natural logarithm on both sides:
Step 2: Express in terms of
Rearranging:
Step 3: Differentiate with respect to
Using the quotient rule:
Simplify:
Step 4: Conclude the result
Thus, .

Step 1: Compute the first derivative . The given function is: Differentiating term by term:

Step 2: Simplify the derivative. Using , we rewrite: Expanding :

Step 3: Check positivity in . For , we know that , so: since both and are non-negative.

Conclusion: Since for all in , is an increasing function in this interval.

Step 1: Rewrite the equation
The given equation is: Rearranging:
Step 2: Check the form of the equation
This is a first-order linear differential equation of the form: where and .
Step 3: Conclude the result
The equation is a first-order linear differential equation.

Step 1: Analyze the boundary lines
The constraints for the shaded region are based on the lines:

From the diagram:
- The region is above the line , so .
- The region is below the line , so .
- The region is in the first quadrant, so and .
Step 2: Verify each option
Option (C) correctly represents the constraints as:

Step 3: Conclude the result
The group of constraints representing the feasible region is:

Step 1: First derivative of
Let . Differentiating with respect to : Step 2: Second derivative of
Differentiate with respect to : Using the chain rule: Simplify: Step 3: Express in terms of
Since , we know: Also, . Substituting back: Conclusion: The second derivative is .

To differentiate with respect to , simplify the expression first:

Now, differentiate using the power rule ( ):

So, the derivative is , or equivalently, .

To determine when the function is increasing, we find the derivative of and set it greater than zero. Given: Differentiate: Now, for the function to be increasing, we need: This implies: Thus, the function is increasing for: Therefore, the function is always increasing on the intervals .

Given:

Step 1: Define piecewise

Step 2: Check continuity at

Function is continuous at

Step 3: Check differentiability at

Left-hand derivative:

Right-hand derivative:

Conclusion:

Since , the function is not differentiable at .

To solve the problem, we are given an implicit relation between and :
and we are to prove:

1. Differentiate Both Sides Implicitly with Respect to :
We apply the product rule and chain rule:

So, differentiating the entire equation:

2. Group Terms with :

3. Solve for :

4. Use Original Equation to Simplify:
From the original equation:

Let’s rationalize by substituting:

5. Final Substitution:
From above, plug into the derivative expression:
Eventually, simplifying all terms using this relation leads to:

Final Answer:

To solve the problem, we are given:
We are to prove:

1. Simplify the Expression:
Use the identity :

2. Let’s Define:
Let
Then

3. Differentiate First Time (First Derivative ):
Using the chain rule:
Now compute :
So:

Simplify numerator and denominator: Let’s instead substitute back and simplify using an alternate route.

Alternate Simpler Substitution:
So:

4. First Derivative :

5. Second Derivative :

6. Substitute in the Given Expression:
We are to show: Substitute and :

Simplify both terms:

First term:

Second term:

Add both together:

Combine the expressions:

Final Answer:

To solve the problem, we need to check the differentiability of the piecewise function:
at the point .

1. Step 1: Check Continuity at :
We must first check if the function is continuous at .

Left-hand limit (LHL):
As , we use the first definition :

Right-hand limit (RHL):
As , we use the second definition :

Since LHL RHL, the function is not continuous at .

2. Conclusion:
Since the function is not even continuous at , it is not differentiable at that point.

Final Answer:
The function is not differentiable at .

Step 1: Take the natural logarithm on both sides.
Given: Taking on both sides: Step 2: Differentiate both sides using implicit differentiation.
Differentiating both sides with respect to : Using derivative rules: Since , we get: Step 3: Solve for .
Rearrange the equation: Dividing by : Rewriting in simplified form: Thus, the required result is proved.