Q1a. Find the derivative of using the limit definition.

Background

Topic: Derivative via Limit Definition

This question tests your understanding of the formal (limit) definition of the derivative, which is foundational in calculus.

Key Terms and Formulas

• Derivative (limit definition):

Step-by-Step Guidance

1. Write the limit definition for using .

2. Compute by substituting into the function: .

3. Expand and subtract : .

4. Simplify the numerator and set up the limit expression over .

Try solving on your own before revealing the answer!

Q1b. Find the derivative of using the limit definition.

Background

Topic: Derivative via Limit Definition

This question reinforces the use of the limit definition for functions involving both rational and polynomial terms.

Key Terms and Formulas

• Derivative (limit definition):

Step-by-Step Guidance

1. Write .

2. Set up the difference quotient: .

3. Combine like terms and simplify the numerator as much as possible.

4. Prepare to take the limit as .

Try solving on your own before revealing the answer!

Q1c. Find the derivative of using the limit definition.

Background

Topic: Derivative via Limit Definition (Radical Functions)

This question tests your ability to apply the limit definition to a function involving a square root in the denominator.

Key Terms and Formulas

• Derivative (limit definition):

Step-by-Step Guidance

1. Write .

2. Set up the difference quotient: .

3. Combine the fractions in the numerator over a common denominator.

4. Simplify the expression and prepare to take the limit as .

Try solving on your own before revealing the answer!

Q1d. Find the derivative of using the limit definition.

Background

Topic: Derivative via Limit Definition (Power Functions)

This question asks you to apply the limit definition to a function with a fractional exponent.

Key Terms and Formulas

• Derivative (limit definition):

Step-by-Step Guidance

1. Write the difference quotient for as above.

2. Consider rationalizing or factoring the numerator to simplify the expression.

3. Set up the limit as and prepare to evaluate.

Try solving on your own before revealing the answer!

Q2a. Compute the derivative of using differentiation rules.

Background

Topic: Differentiation Rules (Sum, Power, Exponential, Trig)

This question tests your ability to apply basic differentiation rules to a function involving polynomials, exponentials, and trigonometric functions.

Key Terms and Formulas

• Power Rule:

• Exponential Rule:

• Trig Rules: ,

Step-by-Step Guidance

1. Differentiate each term separately using the appropriate rule.

2. For , apply the power rule.

3. For , use the exponential rule.

4. For and , use the trigonometric rules.

5. Combine all the derivatives to write .

Try solving on your own before revealing the answer!

Q2b. Compute the derivative of using differentiation rules.

Background

Topic: Product Rule

This question tests your ability to use the product rule for differentiation.

Key Terms and Formulas

• Product Rule:

• Derivatives: ,

Step-by-Step Guidance

1. Let and .

2. Compute and .

3. Apply the product rule formula.

4. Simplify the resulting expression for .

Try solving on your own before revealing the answer!

Q2c. Compute the derivative of using differentiation rules.

Background

Topic: Quotient Rule

This question tests your ability to use the quotient rule for differentiation, involving logarithmic and exponential functions.

Key Terms and Formulas

• Quotient Rule:

• Derivatives: ,

Step-by-Step Guidance

1. Let and .

2. Compute and .

3. Apply the quotient rule formula.

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

Try solving on your own before revealing the answer!

Q2d. Compute the derivative of using differentiation rules.

Background

Topic: Chain Rule

This question tests your ability to use the chain rule for composite functions involving trigonometric and exponential functions.

Key Terms and Formulas

• Chain Rule:

• Derivative:

Step-by-Step Guidance

1. Let , so .

2. Compute .

3. Find .

4. Combine the results to write in terms of .

Try solving on your own before revealing the answer!