Module 3 Practice Problems: Differentiation Rules
Instructions: Work through each problem on paper first, then reveal the solution to check your work. These problems cover the Product Rule, Quotient Rule, Chain Rule, trig derivatives, and implicit differentiation.
Problem 1
Find d/dx [(x³ + 1)(2x − 5)].
Solution
Product Rule: (3x²)(2x − 5) + (x³ + 1)(2) = 6x³ − 15x² + 2x³ + 2 = 8x³ − 15x² + 2.
Problem 2
Find d/dx [(x² + 3x)/(x − 1)].
Solution
Quotient Rule: [(2x + 3)(x − 1) − (x² + 3x)(1)] / (x − 1)²
= [2x² + x − 3 − x² − 3x] / (x − 1)² = (x² − 2x − 3) / (x − 1)².
Problem 3
Find d/dx [(4x − 1)6].
Solution
Chain Rule: 6(4x − 1)5 · 4 = 24(4x − 1)5.
Problem 4
Find d/dx [sin(x² + 1)].
Solution
Chain Rule: cos(x² + 1) · 2x = 2x cos(x² + 1).
Problem 5
Find d/dx [e3x sin x].
Solution
Product Rule + Chain Rule: 3e3x sin x + e3x cos x = e3x(3 sin x + cos x).
Problem 6
Find d/dx [tan(3x²)].
Solution
Chain Rule: sec²(3x²) · 6x = 6x sec²(3x²).
Problem 7
Find d/dx [√(1 + cos x)].
Solution
Rewrite as (1 + cos x)1/2. Chain Rule: (1/2)(1 + cos x)−1/2 · (−sin x) = −sin x / [2√(1 + cos x)].
Problem 8
Find dy/dx for x² + xy + y² = 7.
Solution
Differentiate: 2x + y + x(dy/dx) + 2y(dy/dx) = 0.
(x + 2y)(dy/dx) = −2x − y. So dy/dx = −(2x + y) / (x + 2y).
Problem 9
Find d/dx [ln(sin x)].
Solution
Chain Rule: [1/sin x] · cos x = cos x / sin x = cot x.
Problem 10
Find dy/dx for sin(xy) = x.
Solution
Chain Rule on left: cos(xy) · (y + x dy/dx) = 1.
y cos(xy) + x cos(xy)(dy/dx) = 1.
x cos(xy)(dy/dx) = 1 − y cos(xy).
dy/dx = [1 − y cos(xy)] / [x cos(xy)].