Differentiation

Step 1: Differentiating the function.
The derivative of is: Step 2: Substituting the value of .
At , we know that: So, Step 3: Conclusion.
Thus, the value of at is .

Step 1: Differentiating the function.
The derivative of with respect to is: Step 2: Conclusion.
Thus, the derivative of is 1, corresponding to option (C).

Step 1: Differentiating the function.
We are given that . The first derivative is: Step 2: Finding the second derivative.
Differentiating again, we get: Step 3: Conclusion.
Thus, the second derivative of is , corresponding to option (a).

Step 1: Finding and .
We are given: To find , we first differentiate and with respect to . For , use the quotient rule: Simplifying: So: For , use the quotient rule: Simplifying: Step 2: Finding .
Now, we find by dividing by : Step 3: Verifying the given expression.
We are asked to prove that: Substitute and into the equation: Simplifying: Factor out : Simplifying the expression inside the parentheses: Thus: This simplifies to 0, proving that:

Step 1: Differentiating the first term .
The first term is , which is of the form . To differentiate it, we apply logarithmic differentiation. Take the natural logarithm of both sides: Using the logarithm power rule: Now, differentiate both sides with respect to : Using the product rule for differentiation: Thus: So: Step 2: Differentiating the second term .
Now consider the second term, . Again, we use logarithmic differentiation. Let . Taking the natural logarithm of both sides: Now, differentiate both sides: Using the product rule: Thus: So: Step 3: Final expression for .
Thus, the derivative of is: