Differentiation

Step 1: Use limit and derivative. We use the fact that is the derivative of . The given expression can be rewritten as a difference quotient, which represents the derivative at .
Step 2: Conclusion. Thus, the value of the given expression is 2.

Step 1: Compute partial derivatives. We differentiate the given function with respect to both and .
Step 2: Conclusion. After performing the differentiation, we obtain .

Step 1: Differentiate the given equation.
Differentiate with respect to and using implicit differentiation. Then solve for .

Step 2: Conclusion.
Thus, the correct answer is option (D).

Final Answer:

Step 1: Understanding the relationship between slope and angle.
The slope of the normal is the negative reciprocal of the slope of the tangent, and the slope of the normal is given by . The slope of the tangent is then .

Step 2: Conclusion.
Thus, the correct value of is 1, so the correct answer is option (C).

Final Answer:

Step 1: Recognize standard recurrence relation.
For functions of the form , derivatives satisfy a known recurrence:

Step 2: Use this identity directly.
The given expression matches the LHS.
Hence RHS equals:

Final Answer:

Step 1: Differentiate given expression.

Now:


So total derivative:

Step 2: Write RHS using partial fractions.

Assume:

Multiply both sides by :


Compare coefficients:


Substitute :


Thus:

Step 3: Compare with derivative.

So:

Step 4: Compute .

Final Answer:

Step 1: Simplify expressions using trigonometric interpretation.
Let:

Then:

Step 2: Compare with .

Step 3: Differentiate.
If , then:

Final Answer:

Step 1: Use repeating nature of expression.
Given:
The expression after first repeats itself, so:
But more accurately:
Let inner part after first root be itself:
Step 2: Square both sides.
Step 3: Differentiate implicitly.
Differentiate w.r.t. :
Step 4: Compare with given options.
None of the options matches .
Final Answer:

Step 1: Simplify .
Step 2: Find .
Differentiate:
Step 3: Compute .
Differentiate:
Step 4: Use .
After simplification (standard result for this parametric pair), we get:
Final Answer:

Step 1: Analyzing the derivative.
Given that , the solution function that satisfies the derivative and the given values is . The function satisfies and .

Step 2: Conclusion.
The correct function is , corresponding to option (3).

Step 1: Using the Tangent Formula.
The slope of the tangent to the curve at a point is equal to the derivative of the function at that point. The angle the tangent makes with the positive x-axis is related to the slope by: For , we get . Therefore, .
Step 2: Conclusion.
The correct answer is (A), .

Step 1: Apply the Mean Value Theorem.
For the function , the Mean Value Theorem applies when the derivative of the function equals the difference in function values divided by the interval length.
Step 2: Conclusion.
Thus, the correct value of is .
Final Answer:

Using the sum rule for inverse tangents and the chain rule for differentiation, we find that .
Final Answer:

Differentiate implicitly with respect to and solve for using standard differentiation techniques.
Final Answer:

To solve the inequality, first break it into cases based on the absolute value. After solving, we get .

To determine the behavior of , we differentiate it using the quotient rule: Rewriting the numerator: Step 1: Analyze the sign of Since , the numerator is negative for all . The denominator is always positive. Thus, for all , implying that is monotonically decreasing. Conclusion: The function is decreasing in .

Step 1: Write the given function. The given function is: Step 2: Use the power rule to differentiate. The power rule states that the derivative of is . Now differentiate each term of the function: Step 3: Combine the results. The derivative of is: Answer: Therefore, the derivative is .

Step 1: Use the power rule for differentiation
The power rule for differentiation states that the derivative of is .
Step 2: Differentiate each term We are given: Now, differentiate each term:
- The derivative of is ,
- The derivative of is ,
- The derivative of is ,
- The derivative of the constant is .
Thus, the derivative of is: Answer: Therefore, the derivative of is . So, the correct answer is option (1).

Step 1: Recall the power rule for differentiation
The derivative of is given by: Step 2: Differentiate the given function
We are given .
- The derivative of is (using the power rule: ), - The derivative of is ,
- The derivative of the constant is .
So, the derivative of is: Answer: Therefore, the derivative of is .
So, the correct answer is option (1).