Differentiation

To show that a function is strictly increasing, we need to prove that its derivative is positive for all values of . Step 1: Find the derivative of the function : Step 2: Check the sign of : Since is always positive (because for all and the constant 10 is positive), we conclude that for all . Step 3: Conclusion: Since the derivative is always positive, the function is strictly increasing for all values of .

Step 1: Differentiating the equation.
Given , differentiate both sides implicitly with respect to : Step 2: Solving for .
Substitute and solve for , which turns out to be 0. Step 3: Conclusion.
Thus, the correct answer is (B).

Step 1: Let . This implies: Step 2: Square both sides: Step 3: Rearrange the equation: Step 4: Differentiate implicitly with respect to : Step 5: Factor out : Step 6: Solve for : Step 7: Substitute : Step 8: Evaluate at : At , , so . Thus: Thus, at is:

Step 1: Find
Step 2: Evaluate at

Step 1: Proof of Chain Rule
Since and , we differentiate both functions with respect to : Using the chain rule for derivatives: Step 2: Compute for Given Functions
Given: Differentiate with respect to : Using the chain rule:

Step 1: Take the Logarithm
Taking the natural logarithm on both sides: Step 2: Differentiate Both Sides
Differentiating using implicit differentiation: Using the product rule: Step 3: Solve for
Substituting :

Step 1: Find Slopes of the Lines
The given equation represents two straight lines passing through the origin. The slopes are given by: Step 2: Use the Angle Formula
Substituting values: Step 3: Condition for Coincidence
For coincident lines, , which means: