Topic Review — Differentiation of Trig Functions and Rules

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This review covers all three lessons: Differentiating sin(x) and cos(x), the Chain Rule for Trig/Exponential/Log Functions, and Product and Quotient Rules for All Function Types.

Review Questions

1. Differentiate y = 5 sin x − 3 cos x.
2. Find d⁸/dx⁸[sin x].
3. Find the equation of the tangent to y = cos x at x = π/3.
4. Find all x ∈ [0, 2π] where the gradient of y = 3 sin x + 3 cos x equals zero.
5. Verify that y = A cos x + B sin x satisfies d²y/dx² + y = 0 for any constants A and B.
6. Differentiate y = cos(5x − π).
7. Differentiate y = e^{cos x}.
8. Differentiate y = ln(x² + 1).
9. Differentiate y = (3x² + 1)⁵.
10. Find the gradient of y = sin²(2x) at x = π/8.
11. Differentiate y = x³ cos x.
12. Differentiate y = e^x / cos x.
13. Differentiate y = x ln x and find the x-value where the gradient equals 1.
14. Differentiate y = cos x / (x² + 1).
15. A particle’s velocity is v(t) = t² e^−t m/s for t ≥ 0. Find the acceleration a(t) and determine when the particle reaches its maximum velocity.