Q1. Use derivative rules to find the derivative of .

Background

Topic: Product Rule, Power Rule, and Derivative of Sums

This question tests your ability to apply the product rule, power rule, and sum rule for derivatives to a function that is a product of two expressions.

Key Terms and Formulas

• Product Rule:

• Power Rule:

• Sum Rule:

Step-by-Step Guidance

1. Rewrite as to make differentiation easier.

2. Let and .

3. Find using the power rule: .

4. Find by differentiating each term: and .

5. Set up the product rule: , but do not expand or simplify yet.

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Q2. Find an equation of the line tangent to at .

Background

Topic: Tangent Lines, Product Rule, Trigonometric Derivatives

This question tests your ability to find the equation of a tangent line to a function at a specific point, which involves finding the derivative and evaluating it at a given -value.

Key Terms and Formulas

• Tangent Line Equation:

• Product Rule:

• Derivative of :

Step-by-Step Guidance

1. Let and , then apply the product rule to find .

2. Compute and : , .

3. Plug into to find the -coordinate of the point of tangency.

4. Plug into to find the slope of the tangent line.

5. Set up the tangent line equation using the point and the slope, but do not simplify or solve for yet.

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Q3. Suppose and . Let . Find the equation of the tangent line to at .

Background

Topic: Chain Rule, Product Rule, Tangent Lines

This question tests your ability to use the product and chain rules to differentiate a composite function, and then use that derivative to find a tangent line.

Key Terms and Formulas

• Product Rule:

• Chain Rule:

• Tangent Line Equation:

Step-by-Step Guidance

1. Let and , then apply the product rule to find .

2. Use the chain rule to differentiate : .

3. Evaluate using the given values for .

4. Evaluate using the given values for and .

5. Set up the tangent line equation at , but do not simplify or solve for yet.

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Q4. Use the table to find the following:

Background

Topic: Derivatives from Tables, Product Rule, Tangent Lines

This question tests your ability to use tabular data to compute derivatives and tangent lines for functions defined in terms of and .

Key Terms and Formulas

• Product Rule:

• Tangent Line Equation:

Step-by-Step Guidance

1. For (a), use the constant multiple rule: , then evaluate at using the table.

2. For (b), use the product rule for , then substitute and use the table values for and .

3. For (c), if , use the quotient rule to find and evaluate at using the table.

4. Set up the tangent line equation for (c) using the values found, but do not simplify or solve for yet.

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Q5. Find the derivatives of the following functions:

Background

Topic: Basic Derivative Rules, Chain Rule, Trigonometric and Logarithmic Derivatives

This question tests your ability to apply the chain rule, power rule, and derivatives of trigonometric and logarithmic functions.

Key Terms and Formulas

• Derivative of :

• Chain Rule:

• Derivative of : Use chain rule with

• Derivative of :

Step-by-Step Guidance

1. For each part, identify the outer and inner functions and apply the appropriate rule (chain, product, etc.).

2. For (a), rewrite as and use the chain rule.

3. For (b), recognize as and use the chain rule.

4. For (c), apply the chain rule to .

5. For (d), use the chain rule for .

6. For (e), apply the chain rule to .

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Q6. Implicitly differentiate the following equations to find :

Background

Topic: Implicit Differentiation

This question tests your ability to use implicit differentiation to find when is not isolated.

Key Terms and Formulas

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

• Product Rule and Chain Rule may be needed.

Step-by-Step Guidance

1. For (a), differentiate both sides of with respect to , remembering to use the product rule on and chain rule on .

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

3. Factor out and set up the equation to solve for it, but do not isolate yet.

4. For (b), differentiate both sides of with respect to , using the chain rule for and product rule as needed.

5. Collect and factor as above, but do not solve for yet.

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Q7. For the implicitly defined curve , find .

Background

Topic: Second Derivative, Implicit Differentiation

This question tests your ability to find the second derivative of with respect to for an implicitly defined curve.

Key Terms and Formulas

• First, find using implicit differentiation.

• Then, differentiate with respect to again, using the chain rule as needed.

Step-by-Step Guidance

1. Differentiate both sides of with respect to to find .

2. Express in terms of and .

3. Differentiate with respect to again, using the chain rule for as a function of .

4. Set up the expression for , but do not simplify or solve for it yet.

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Q8. Find the derivative of .

Background

Topic: Product Rule, Trigonometric Derivatives

This question tests your ability to use the product rule and derivatives of trigonometric functions.

Key Terms and Formulas

• Product Rule:

Step-by-Step Guidance

1. Let and , then apply the product rule.

2. Find and using the trigonometric derivatives above.

3. Set up the expression for using the product rule, but do not simplify yet.

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Q9. Suppose the position (in feet) of a football thrown upwards vertically at any time (seconds) is . Find the velocity and acceleration functions.

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:

Step-by-Step Guidance

1. Differentiate with respect to to find .

2. Differentiate with respect to to find .

3. Write the expressions for and , but do not simplify or evaluate them yet.

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Q10. A 15-foot ladder leans against a vertical wall. The bottom of the ladder is sliding away from the wall at a rate of 2 feet per second.

Background

Topic: Related Rates

This question tests your ability to use related rates to solve problems involving changing quantities in geometric situations.

Key Terms and Formulas

• Pythagorean Theorem:

• Related Rates: Differentiate both sides with respect to to relate and .

• Angle Rate: is the angle between the ladder and the ground.

Step-by-Step Guidance

1. Let be the distance from the wall to the bottom of the ladder, be the height on the wall, and the angle with the ground.

2. Differentiate with respect to to relate and .

3. Plug in and to set up the equation for , but do not solve for yet.

4. For the angle, use and differentiate with respect to to relate to .

5. Set up the equation for , but do not solve for it yet.

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Q11. Air is being pumped into a spherical balloon at a rate of 100 cm/s. How fast is the diameter of the balloon increasing when the radius is 5 cm?

Background

Topic: Related Rates, Volume of a Sphere

This question tests your ability to relate the rate of change of volume to the rate of change of diameter using related rates.

Key Terms and Formulas

• Volume of a sphere:

• Diameter:

• Related Rates:

Step-by-Step Guidance

1. Differentiate with respect to to relate and .

2. Plug in and to set up the equation for , but do not solve for yet.

3. Since , ; set up this relationship but do not solve for yet.

Try solving on your own before revealing the answer!