Differentiability

Given, } \\[2ex]
k, & \text{for }
\end{cases}\)

Given,
on differentiating both sides w.r.t. , we get

[by using product rule of derivative]


We know that, if Rolle's theorem is verified,
then their exist , such that

Let and
On differentiating both sides w.r.t. we get

and

Answer (c)

we have given integral is: To simplifying the integral, rewrite it as: Next, perform a substitution: Let Under this substitution, the following transformations occur: This changes the integral to: Let this new integral be denoted as . Now, by adding the original integral and the substituted integral , we get: Simplifying the right-hand side: The limits of integration give: Simplifying this result: Thus, . Since , we have: Solving for : Final Answer:

The given logical expression is: Step 1: Simplify the expression. Applying Boolean algebra, we have: Thus, the expression becomes: Step 2: Rewrite in implication form. Recognizing that: we conclude that the simplified expression is equivalent to . Therefore, the correct answer is .

Step 1: Use the power rule for differentiation The power rule states that if , then . Step 2: Differentiate each term of the function The function is . Let's differentiate each term: 1. The derivative of is: 2. The derivative of is: 3. The derivative of the constant is: Step 3: Combine the results The derivative of is: Answer: Therefore, the derivative of the function is . So, the correct answer is option (1).