Differentiation

Differentiate termwise: , . Add them.

. Using chain rule on : derivative . Add to get .

First derivative: . Second derivative: differentiate again .

Idea. Derivative of is . Here so .

Idea. For base (constant), . and (base ) are the same in calculus notation here.
So .

Idea. Write as a negative power and apply the chain rule.

Idea. Use parametric differentiation: . Trig double-angle simplifies nicely.
Step 1.
Step 2.
Step 3. .

; ; . Sum them to get .

. Constant's derivative is . . Hence the result is .

Step 1: Differentiate the function with respect to to get the slope of the tangent. Step 2: Evaluate the derivative at : - : This is defined only if . If , the term becomes 2. If , the derivative diverges. Assuming (most plausible integer for the given form), then: At : But the correct answer is marked as (C) , which suggests either: - : then , derivative of constant is 0 - : then derivative = 2 - Best assumption for the slope to be 0 is , but that leads to slope 2 + 3 = 5. So to get slope 0, perhaps and original function is: Then, None of these lead to 0 unless and a cancellation occurs. We must assume that , as initially tried, then: So Answer: (A) 3 (Seems like there was an error in the source you gave — you can correct the answer or clarify the value of .)

Step 1: The area of a circle in terms of radius is: Step 2: To find the rate of change of area with respect to radius, differentiate with respect to : Step 3: Substitute cm: Final Answer:

Given: Take natural logarithm on both sides: Differentiate both sides with respect to , treating as a function of : Using product rule on both sides: Rewrite: Group terms on one side: Factor : Therefore,

We need to differentiate with respect to . Using the chain rule: This simplifies to: which is the correct result. Therefore, the correct option is:

Differentiate both sides with respect to , using implicit differentiation: Using product and chain rules on the left: Rearranged: Group terms involving : Factor : Therefore,

Let Notice that since Also, but more simply, because and . Therefore, Hence,
Final answer:

Differentiate and with respect to : Using the chain rule for parametric functions:
Final answer:

Rewrite using logarithms: Differentiate both sides with respect to : Multiply both sides by : Substitute back :

Rewrite: Using the chain rule: Differentiate inside: Therefore,

We are given: Let: Now square both sides: Differentiate both sides with respect to :

Given: Now use the chain rule: Wait — let's correct this: So: This matches Option (B), not (A). Correct Answer: (B)
Correct Answer Correct Answer: (B)

We need to differentiate with respect to . Using basic differentiation rules: Next, applying the chain rule to : Thus, the derivative is: Therefore, the correct option is:

To differentiate , we use the standard formula for the derivative of the inverse secant function: Thus, the derivative is: Therefore, the correct option is:

To differentiate an exponential function with base (where and ), we use the formula: Hence, the derivative of is:

To differentiate , we use the chain rule. The derivative of is , and then we differentiate to get . Thus: This simplifies to: Therefore, the correct option is:

We are given the function . To find the derivative, we apply the chain rule. First, let: The derivative of with respect to is: Now, we find the derivative of using the product rule and chain rule: Differentiating with respect to , we get: Thus, the derivative of becomes: Now applying the chain rule: Simplifying the terms, we find the correct answer is: Thus, the correct answer is option (C).

We are given the function , which can be written as . Taking the derivative of , we get: Thus, the correct answer is option (B) .

We are asked to differentiate . Using the chain rule for differentiation, where the derivative of with respect to is , we have: The derivative of with respect to is , so the final result is: Thus, the correct answer is option (C).

We apply the quotient rule for differentiation. The quotient rule states: Where and . Step 1: Differentiate We use the product rule for differentiating , which is: Where and . So, Simplifying, Step 2: Differentiate Since , Step 3: Apply the quotient rule Now, applying the quotient rule: Upon simplification, you can see that the final answer for this derivative simplifies to 0, which corresponds to option D. Thus, the correct answer is:

We are asked to differentiate with respect to . Using the chain rule: Since is a constant, its derivative is zero. Thus: But because we have and not just , we must apply the chain rule: Thus, the correct answer is .