Derivatives of e^x, ln x, trig; chain/product/quotient rules practice questions.

Determine the derivative of y = sin x / x for x ≠ 0.

• (sin x − x cos x)/x²
• cos x
• (x cos x + sin x)/x²
• (x cos x − sin x)/x²

Answer: (x cos x − sin x)/x²

By the quotient rule, dy/dx = (x·cos x − sin x·1)/x². Reversing the order of the terms in the numerator is the standard sign error.

Determine h′(x) given h(x) = cos(3x).

• −3 sin(3x)
• 3 sin(3x)
• −sin(3x)
• −3 cos(3x)

Answer: −3 sin(3x)

By the chain rule, differentiate the outer function (cos → −sin) and multiply by the derivative of the inner function 3x, giving −3 sin(3x).

Determine the derivative of y = e^(x²).

• e^(x²)
• x²·e^(x² − 1)
• 2x·e^(x²)
• 2x·e^(2x)

Answer: 2x·e^(x²)

By the chain rule, d/dx e^(x²) = e^(x²) × d/dx(x²) = 2x·e^(x²). Treating the exponential like a power of x gives the distractor x²·e^(x²−1).

Determine dy/dx given y = x² ln x.

• 2x ln x
• 2x ln x + x
• 2x/x
• 2x ln x + x²

Answer: 2x ln x + x

By the product rule, dy/dx = 2x·ln x + x²·(1/x) = 2x ln x + x. Dropping the second product-rule term gives 2x ln x; taking the derivative of ln x as 1 instead of 1/x gives 2x ln x + x².

Determine g′(x) given g(x) = (4 − x²)³.

• 3(4 − x²)²
• −2x(4 − x²)²
• 6x(4 − x²)²
• −6x(4 − x²)²

Answer: −6x(4 − x²)²

The chain rule gives 3(4 − x²)² multiplied by the inner derivative −2x, so g′(x) = −6x(4 − x²)². Omitting the inner derivative or its minus sign gives the distractors.

Determine the derivative of y = (x + 1)/(x − 1).

• -2/(x - 1)²
• 2/(x - 1)²
• -2/(x + 1)²
• -1/(x - 1)²

Answer: -2/(x - 1)²

Quotient rule: y′ = [1·(x − 1) − (x + 1)·1]/(x − 1)² = (x − 1 − x − 1)/(x − 1)² = -2/(x − 1)². Getting the numerator subtraction order wrong gives +2/(x − 1)².