Q1. Find the first derivative of

Background

Topic: Differentiation – Basic Rules

This question tests your ability to find the derivative of a function involving powers, exponentials, and trigonometric functions. You may need to use the sum rule and basic derivatives.

Key Terms and Formulas:

• Power Rule:

• Derivative of with respect to :

• Derivative of with respect to :

• Sum Rule: The derivative of a sum is the sum of the derivatives.

Step-by-Step Guidance

1. Differentiate each term separately using the sum rule.

2. Apply the power rule to .

3. Differentiate with respect to .

4. Differentiate with respect to .

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Q2. Find the first derivative of

Background

Topic: Product Rule

This question tests your ability to use the product rule to differentiate a product of two functions.

Key Terms and Formulas:

• Product Rule:

• Derivative of :

• Derivative of :

Step-by-Step Guidance

1. Let and .

2. Find and .

3. Apply the product rule formula.

4. Combine like terms if possible.

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Q3. Find the first derivative of

Background

Topic: Product Rule and Trigonometric Derivatives

This question tests your ability to differentiate a product involving a trigonometric function and a variable.

Key Terms and Formulas:

• Product Rule:

• Derivative of with respect to :

• Derivative of with respect to :

Step-by-Step Guidance

1. Identify and .

2. Find the derivatives and .

3. Apply the product rule to combine the derivatives.

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Q4. Use the quotient rule to find the derivative of

Background

Topic: Quotient Rule

This question tests your ability to use the quotient rule to differentiate a function that is a ratio of two functions.

Key Terms and Formulas:

• Quotient Rule:

• Derivative of :

• Derivative of :

Step-by-Step Guidance

1. Let and .

2. Find and .

3. Apply the quotient rule formula.

4. Simplify the numerator and denominator as much as possible.

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Q5. Find for the implicitly defined curve

Background

Topic: Implicit Differentiation

This question tests your ability to use implicit differentiation to find when is defined implicitly as a function of .

Key Terms and Formulas:

• Implicit Differentiation: Differentiate both sides of the equation with respect to , treating as a function of .

• Chain Rule:

Step-by-Step Guidance

1. Differentiate both sides of the equation with respect to .

2. Apply the chain rule to the term.

3. Collect all terms involving on one side of the equation.

4. Solve for in terms of and .

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Q6. A stone is thrown off the edge of a cliff. Its height above the ground at time is . Find and .

Background

Topic: Derivatives as Velocity and Acceleration

This question tests your understanding of how the first derivative of position gives velocity, and the second derivative gives acceleration.

Key Terms and Formulas:

• Velocity:

• Acceleration:

• Power Rule:

Step-by-Step Guidance

1. Differentiate with respect to to find .

2. Differentiate with respect to to find .

3. Interpret the meaning of the signs of and in the context of the problem.

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Q7. Oil spilled from a ruptured tanker spreads in a circle whose area increases at a constant rate of $6^2 mi?

Background

Topic: Related Rates

This question tests your ability to relate the rates of change of area and radius for a circle using related rates.

Key Terms and Formulas:

• Area of a circle:

• Differentiate both sides with respect to :

Step-by-Step Guidance

1. Write the formula relating area and radius.

2. Differentiate both sides with respect to time to relate and .

3. Plug in the given values for and .

4. Rearrange to solve for .

Try solving on your own before revealing the answer!