Differentiate from first principles

Question 1

For f(x) = x² − 5x, simplify (f(x + h) − f(x))/h.

• A. x² + h − 5
• B. 2xh + h²
• C. 2x − 5
• D. 2x + h − 5

Question 2

Using first principles, f(x) = x² − 2x gives (f(x + h) − f(x))/h = 2x + h − 2. What is f′(x)?

Answer 

Question 3

For f(x) = x² + 5x, simplify (f(x + h) − f(x))/h.

• A. 2x + 5
• B. x² + h + 5
• C. 2x + h + 5
• D. 2x + h² + 5

Question 4

Using first principles, f(x) = −x² + 4x gives (f(x + h) − f(x))/h = −2x − h + 4. What is f′(x)?

Answer 

Question 5

For f(x) = x² + 6x, simplify (f(x + h) − f(x))/h.

• A. 2x + h² + 6
• B. 2x + h + 6
• C. x² + h + 6
• D. 2xh + h²

Question 6

Using first principles, f(x) = 5x² − 3x gives (f(x + h) − f(x))/h = 10x + 5h − 3. What is f′(x)?

Answer 

Question 7

For f(x) = x² − 3x, simplify (f(x + h) − f(x))/h.

• A. x² + h − 3
• B. 2x + h² − 3
• C. 2xh + h²
• D. 2x + h − 3

Question 8

Using first principles, f(x) = −x² gives (f(x + h) − f(x))/h = −2x − h. What is f′(x)?

Answer 

Question 9

For f(x) = x² − 6x, simplify (f(x + h) − f(x))/h.

• A. 2x + h − 6
• B. x² + h − 6
• C. 2x + h² − 6
• D. 2xh + h²

Question 10

Using first principles, f(x) = 3x² − 3x gives (f(x + h) − f(x))/h = 6x + 3h − 3. What is f′(x)?

Answer 

Question 11

For f(x) = x² + 3x, simplify (f(x + h) − f(x))/h.

• A. x² + h + 3
• B. 2x + h + 3
• C. 2x + h² + 3
• D. 2x + 3

Question 12

Using first principles, f(x) = −5x² + 5x gives (f(x + h) − f(x))/h = −10x − 5h + 5. What is f′(x)?

Answer 

Question 13

For f(x) = x² + 3x, simplify (f(x + h) − f(x))/h.

• A. 2x + h² + 3
• B. 2x + 3
• C. x² + h + 3
• D. 2x + h + 3

Question 14

Using first principles, f(x) = −3x² + 5x gives (f(x + h) − f(x))/h = −6x − 3h + 5. What is f′(x)?

Answer 

Question 15

For f(x) = x² − 3x, simplify (f(x + h) − f(x))/h.

• A. x² + h − 3
• B. 2x − 3
• C. 2x + h² − 3
• D. 2x + h − 3

Question 16

Using first principles, f(x) = −3x² − 6x gives (f(x + h) − f(x))/h = −6x − 3h − 6. What is f′(x)?

Answer