Power rule; derivatives of polynomials practice questions.

Let f(x) = 3x⁴ − 2x² + 7. Determine f′(x).

• 12x³ − 4x + 7
• 12x³ − 4x
• 3x³ − 2x
• 12x⁴ − 4x²

Answer: 12x³ − 4x

Differentiate term by term with the power rule: d/dx(3x⁴) = 12x³, d/dx(−2x²) = −4x, and the constant 7 differentiates to 0.

The curve y = 1/x² can be written as y = x⁻². Its derivative dy/dx is

• 2/x³
• −2x
• −2/x
• −2/x³

Answer: −2/x³

Apply the power rule with n = −2: dy/dx = −2x⁻³ = −2/x³. The exponent decreases from −2 to −3, and the sign is negative.

Let f(x) = 2x³ − 5x² + 4x − 1. Calculate f′(1).

• 0
• 6
• 4
• −4

Answer: 0

f′(x) = 6x² − 10x + 4, so f′(1) = 6 − 10 + 4 = 0. The curve has a horizontal gradient at x = 1.

Let y = √x. Determine dy/dx.

• 2√x
• √x/2
• 1/(2√x)
• −1/(2√x)

Answer: 1/(2√x)

√x = x^(1/2), so dy/dx = (1/2)x^(−1/2) = 1/(2√x). The power rule works for rational exponents too.

Let f(x) = (2x + 1)(x − 3). Determine f′(x).

• 4x − 5
• 2x − 5
• 2
• 4x − 6

Answer: 4x − 5

Expand first: f(x) = 2x² − 5x − 3, so f′(x) = 4x − 5. Multiplying the two derivatives (giving 2) is a classic error.

Let f(x) = x⁵/5 − x³/3 + x. Determine f′(x).

• 5x⁴ − 3x² + 1
• x⁴ − x²
• x⁴ − x² + 1
• x⁴ − x² + x

Answer: x⁴ − x² + 1

The coefficients 1/5 and 1/3 cancel with the powers: d/dx(x⁵/5) = x⁴, d/dx(−x³/3) = −x², and d/dx(x) = 1.