Differentiation: Definition and Basic Rules — High School Calculus practice questions

Find d/dx[1/x²].

1. −2/x³
2. 2x
3. −2/x
4. 1/(2x)

Answer: A — −2/x³

A) Correct — 1/x² = x^(−2), so the derivative is −2x^(−3). B) Treats 1/x² as if it were x². C) Adds 1 to the exponent instead of subtracting 1. D) Differentiates only the denominator.

An object's position is s(t) = t² + 3t. What is its velocity v(t)?

1. t + 3
2. 2t
3. t³/3 + 3t²/2
4. 2t + 3

Answer: D — 2t + 3

D) Correct — Velocity is the derivative of position. A) Forgets the coefficient 2 from the power rule. B) Drops the derivative of the 3t term. C) Integrates instead of differentiating.

What is the slope of the tangent line to y = x³ at x = 2?

1. 8
2. 6
3. 12
4. 4

Answer: C — 12

C) Correct — y' = 3x², and 3(2)² = 12. A) Is the y-value, 2³, not the slope. B) Uses 3x instead of 3x². D) Uses 2x, the derivative of x².

For f(x) = x² on [1, 3], what is the average rate of change?

1. 6
2. 2
3. 8
4. 4

Answer: D — 4

D) Correct — [f(3) − f(1)]/(3 − 1) = (9 − 1)/2 = 4. A) Is the instantaneous rate f'(3). B) Is the instantaneous rate f'(1). C) Finds f(3) − f(1) but forgets to divide by 2.

Which limit defines the derivative f'(x)?

1. lim_{h→0} [f(x + h) − f(x)]/h
2. lim_{h→0} [f(x + h) − f(x)]
3. [f(b) − f(a)]/(b − a)
4. lim_{h→0} f(x + h)/h

Answer: A — lim_{h→0} [f(x + h) − f(x)]/h

A) Correct — The derivative is the limit of the difference quotient. B) Leaves out the division by h; this limit is 0 for continuous f. C) Is the average rate of change, with no limit taken. D) Leaves out the subtraction of f(x).

Find d/dx[x eˣ].

1. eˣ
2. x eˣ
3. (x + 1)eˣ
4. (x − 1)eˣ

Answer: C — (x + 1)eˣ

C) Correct — Product rule: 1·eˣ + x·eˣ = (x + 1)eˣ. A) Multiplies the derivatives: 1·eˣ. B) Differentiates only the eˣ factor. D) Subtracts the two product-rule terms instead of adding.

Find d/dx[4x³ − 5x + 2].

1. 12x² − 5x
2. 12x² − 5
3. 12x² − 5 + 2
4. 4x² − 5

Answer: B — 12x² − 5

B) Correct — Power rule term by term: 12x², −5, and 0 for the constant. A) Keeps the x on the linear term; d/dx[5x] = 5. C) Forgets that the derivative of a constant is 0. D) Lowers the exponent but forgets to multiply by it.

What is the power rule?

d/dx[xⁿ] = n·xⁿ⁻¹.

What is the limit definition of the derivative?

f'(x) = lim_{h→0} [f(x + h) − f(x)]/h.

What is the quotient rule?

(f/g)' = (f'g − fg')/g².

Find d/dx[(2x + 1)³] using chain rule.

3(2x + 1)² · 2 = 6(2x + 1)².