Fundamental Theorem Of Calculus

Step 1: Understanding the Concept:
To find the derivative , we use the power rule of differentiation: .
Step 2: Detailed Explanation:
Apply the rule to each term of the function :

• For : The derivative is .
• For : The derivative is .

Combining these, we get .
Step 3: Final Answer:
The correct option is (a).

Step 1: Understanding the Concept:
Integration is the reverse process of differentiation. For a power function , the power rule for integration is: (where ).
Step 2: Detailed Explanation:
In this question, can be written as . Applying the rule:
$ $
Step 3: Final Answer:
The correct option is (b).

Step 1: Understanding the Concept:
Standard trigonometric derivatives are fundamental formulas in calculus. The derivative represents the rate of change or the slope of the tangent to the curve at any point.
Step 2: Detailed Explanation:
By using the first principle of derivatives or standard limits: $ $
Step 3: Final Answer:
The correct option is (a).