Module 3 Quiz: Differentiation Rules
Quiz Instructions
Answer each question, then reveal the answer to check your work. Aim for at least 7 out of 10 correct.
1. Find d/dx [x² ex].
Product Rule: 2x ex + x² ex = ex(x² + 2x).
2. Find d/dx [(5x + 2)/(x² − 1)].
Quotient Rule: [5(x² − 1) − (5x + 2)(2x)] / (x² − 1)² = [5x² − 5 − 10x² − 4x] / (x² − 1)² = (−5x² − 4x − 5) / (x² − 1)².
3. Find d/dx [(3x² + 2)4].
Chain Rule: 4(3x² + 2)³ · 6x = 24x(3x² + 2)³.
4. Find d/dx [cos(5x)].
Chain Rule: −sin(5x) · 5 = −5 sin(5x).
5. Find d/dx [sec x · tan x].
Product Rule: (sec x tan x)(tan x) + (sec x)(sec² x) = sec x tan² x + sec³ x = sec x (tan² x + sec² x).
6. Find d/dx [esin x].
Chain Rule: esin x · cos x = cos x · esin x.
7. Find d/dx [ln(x² + 4)].
Chain Rule: 2x / (x² + 4) = 2x / (x² + 4).
8. Find dy/dx for x³ + y³ = 9.
3x² + 3y²(dy/dx) = 0. So dy/dx = −x² / y².
9. Find d/dx [sin²(x)].
Chain Rule: 2 sin x · cos x = sin 2x.
10. Find dy/dx for x²y + y³ = 10.
Differentiate: 2xy + x²(dy/dx) + 3y²(dy/dx) = 0. Factor: (x² + 3y²)(dy/dx) = −2xy. So dy/dx = −2xy / (x² + 3y²).