Q1a. Find the derivative of the function:

Background

Topic: Differentiation of Polynomials and Rational Functions

This question tests your ability to apply the power rule, constant multiple rule, and the derivative of negative and fractional exponents.

Key Terms and Formulas

• Power Rule:

• Derivative of a constant times a function:

• Derivative of :

Step-by-Step Guidance

1. Rewrite all terms with negative or fractional exponents: , stays as is, .

2. Apply the power rule to each term separately. For example, .

3. For each term, multiply the coefficient by the new exponent and decrease the exponent by one.

4. Combine all the differentiated terms to form the derivative .

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Q1b. Find the derivative of the function:

Background

Topic: Differentiation of Rational, Exponential, and Trigonometric Functions

This question tests your ability to differentiate terms with negative exponents, fractional exponents, constants, exponentials, and trigonometric functions.

Key Terms and Formulas

• Power Rule:

• Derivative of :

• Derivative of :

Step-by-Step Guidance

1. Rewrite as .

2. Apply the power rule to and .

3. The derivative of a constant (19) is zero.

4. Differentiate and using their respective rules.

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Q1c. Find the derivative of the function:

Background

Topic: Product Rule and Exponential Functions

This question tests your ability to use the product rule when differentiating a product of an exponential and a polynomial.

Key Terms and Formulas

• Product Rule:

• Derivative of :

Step-by-Step Guidance

1. Let and .

2. Compute and separately.

3. Apply the product rule: .

4. Simplify the resulting expression by factoring if possible.

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Q2. Write the slope of the tangent (rate of change) at the point for the function

Background

Topic: Derivatives and Tangent Slopes

This question tests your ability to find the derivative of a function and evaluate it at a specific point to find the slope of the tangent line.

Key Terms and Formulas

• Derivative at a point: gives the slope of the tangent at .

Step-by-Step Guidance

1. First, find the derivative of the given function (as in Q1a).

2. Substitute into the derivative .

3. Evaluate each term of at and sum them to get the slope.

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Q3. At what point(s) is the rate of change zero (slope of the tangent = 0) for the function ?

Background

Topic: Critical Points and Setting Derivative to Zero

This question tests your ability to find where the derivative of a function is zero, which corresponds to horizontal tangents or critical points.

Key Terms and Formulas

• Critical points: Points where .

• Product Rule:

Step-by-Step Guidance

1. Find the derivative using the product rule.

2. Set to find the -values where the slope is zero.

3. Solve the resulting equation for .

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Q11. An ice cube is melting and the volume is decreasing at the rate of . Find the rate at which its side is changing when the side is . How is surface area changing at that point?

Background

Topic: Related Rates (Implicit Differentiation)

This question tests your ability to relate the rates of change of different quantities (volume, side length, surface area) for a cube using differentiation.

Key Terms and Formulas

• Volume of a cube:

• Surface area of a cube:

• Related rates: ,

Step-by-Step Guidance

1. Write the formula for the volume of a cube: .

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

3. Plug in the given values: (negative because the volume is decreasing), .

4. Solve for , the rate at which the side is changing.

5. Use and differentiate with respect to to find in terms of and .

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