Topic Review — Differentiation of Trig Functions and Rules — Solutions

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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. ▶ Show Answer
   dy/dx = 5 cos x + 3 sin x
2. Find d⁸/dx⁸[sin x]. ▶ Show Answer
   8 ÷ 4 = 2 remainder 0. Remainder 0 → the function returns to sin x.
   d⁸/dx⁸[sin x] = sin x
3. Find the equation of the tangent to y = cos x at x = π/3. ▶ Show Answer
   At x = π/3: y = cos(π/3) = 1/2. Point: (π/3, 1/2).
   dy/dx = −sin x; at x = π/3: gradient = −sin(π/3) = −√3/2
   Tangent: y − 1/2 = (−√3/2)(x − π/3)
4. Find all x ∈ [0, 2π] where the gradient of y = 3 sin x + 3 cos x equals zero. ▶ Show Answer
   dy/dx = 3 cos x − 3 sin x = 0 ⇒ cos x = sin x ⇒ tan x = 1
   x = π/4 or x = 5π/4
5. Verify that y = A cos x + B sin x satisfies d²y/dx² + y = 0 for any constants A and B. ▶ Show Answer
   dy/dx = −A sin x + B cos x
   d²y/dx² = −A cos x − B sin x
   d²y/dx² + y = (−A cos x − B sin x) + (A cos x + B sin x) = 0 ✓
6. Differentiate y = cos(5x − π). ▶ Show Answer
   Inner function: 5x − π; inner derivative: 5.
   dy/dx = −5 sin(5x − π)
7. Differentiate y = e^{cos x}. ▶ Show Answer
   Outer: e^{( )}; inner: cos x; inner derivative: −sin x
   dy/dx = −sin x · e^{cos x}
8. Differentiate y = ln(x² + 1). ▶ Show Answer
   Outer: ln( ); inner: x² + 1; inner derivative: 2x
   dy/dx = 2x / (x² + 1)
9. Differentiate y = (3x² + 1)⁵. ▶ Show Answer
   Outer: ( )⁵; inner: 3x² + 1; inner derivative: 6x
   dy/dx = 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴
10. Find the gradient of y = sin²(2x) at x = π/8. ▶ Show Answer
    y = (sin(2x))²
    dy/dx = 2 sin(2x) · 2 cos(2x) = 4 sin(2x) cos(2x) = 2 sin(4x)
    At x = π/8: dy/dx = 2 sin(π/2) = 2 × 1 = 2
11. Differentiate y = x³ cos x. ▶ Show Answer
    u = x³, v = cos x; u′ = 3x², v′ = −sin x
    dy/dx = 3x² cos x − x³ sin x = x²(3 cos x − x sin x)
12. Differentiate y = e^x / cos x. ▶ Show Answer
    u = e^x, v = cos x; u′ = e^x, v′ = −sin x
    dy/dx = (e^x cos x − e^x(−sin x)) / cos²x
    = e^x(cos x + sin x) / cos²x
13. Differentiate y = x ln x and find the x-value where the gradient equals 1. ▶ Show Answer
    u = x, v = ln x; u′ = 1, v′ = 1/x
    dy/dx = ln x + x · (1/x) = ln x + 1
    Gradient = 1: ln x + 1 = 1 ⇒ ln x = 0 ⇒ x = 1
14. Differentiate y = cos x / (x² + 1). ▶ Show Answer
    u = cos x, v = x² + 1; u′ = −sin x, v′ = 2x
    dy/dx = (−sin x · (x² + 1) − cos x · 2x) / (x² + 1)²
    = −[(x² + 1) sin x + 2x cos x] / (x² + 1)²
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. ▶ Show Answer
    v(t) = t² e^−t
    u = t², v = e^−t; u′ = 2t, v′ = −e^−t
    a(t) = v′(t) = 2t e^−t + t²(−e^−t) = e^−t(2t − t²) = te^−t(2 − t)
    a(t) = 0: t = 0 or t = 2. Since v(0) = 0 and v(2) = 4e⁻² > 0, the maximum velocity occurs at t = 2 s.