First principles; higher derivatives practice questions.

In a first-principles differentiation of f(x) = x², the quotient (f(x + h) − f(x)) / h simplifies, before the limit is taken, to:

• 2x
• x + h
• 2x + h²
• 2x + h

Answer: 2x + h

(x + h)² − x² = 2xh + h², and dividing by h gives 2x + h; the limit h → 0 then yields f′(x) = 2x.

When differentiating f(x) = 1/x from first principles, the quotient (f(x + h) − f(x)) / h simplifies, before the limit is taken, to:

• −1 / (x(x + h))
• 1 / (x(x + h))
• −h / (x(x + h))
• −1 / x²

Answer: −1 / (x(x + h))

1/(x + h) − 1/x = −h/(x(x + h)); dividing by h leaves −1/(x(x + h)), which tends to −1/x² as h → 0.

Determine the third derivative f⁽³⁾(x) of f(x) = x⁴.

• 12x²
• 24
• 24x
• 4x³

Answer: 24x

Differentiating repeatedly: f′(x) = 4x³, f″(x) = 12x², f⁽³⁾(x) = 24x; the constant 24 is the fourth derivative.

Let y = e²ˣ. Determine an expression for the nth derivative dⁿy/dxⁿ.

• 2ⁿ e²ˣ
• 2e²ˣ
• e²ˣ
• 2n e²ˣ

Answer: 2ⁿ e²ˣ

Each differentiation multiplies by the factor 2 from the chain rule, so after n differentiations the result is 2ⁿ e²ˣ.

Let f(x) = sin x. Determine f⁽²⁰²⁶⁾(x), the 2026th derivative of f.

• sin x
• cos x
• −cos x
• −sin x

Answer: −sin x

The derivatives of sin x cycle with period 4: cos x, −sin x, −cos x, sin x. Since 2026 = 4 × 506 + 2, the 2026th derivative equals the second, −sin x.

Let y = ln x, x > 0. Determine d²y/dx².

• 1/x²
• −1/x²
• −1/x
• −2/x³

Answer: −1/x²

dy/dx = 1/x = x⁻¹, so d²y/dx² = −x⁻² = −1/x²; forgetting the negative sign from the power rule is a common error.