Let y=f(x)=sin ³(π/3(cos (π/(3 √2)(-4 x³+5 x²+1)^(3/2)))). Then, at x = 1,

Solution

Maths sir · 4 steps

Find every inner layer at x = 1 first; each angle is a standard one, so the chain rule stays easy.

1. Inner values. Let u = -4x³ + 5x² + 1, φ = π/3√2u^(3/2) and θ = π/3 cos φ, so y = sin³ θ. At x = 1: u = 2 and u^(3/2) = 2√2, so φ = 2π/3. Then θ = π/3(-1/2) = -π/6 and y = (-1/2)³ = -1/8.

   y' = 3 sin² θ cos θ × (-π/3 sin φ) × π/3√2·3/2u^(1/2) × u'

2. Inner factors. u' = -12x² + 10x = -2 at x = 1, so φ' = π/3√2 × 3/2 × √2 × (-2) = -π. Then θ' = -π/3 × √3/2 × (-π) = √3π²/6.
3. Outer factor. y' = 3 × 1/4 × √3/2 × √3π²/6 = 9π²/48 = 3π²/16.
4. So. 2y' + 3π² y = 3π²/8 - 3π²/8 = 0, which is option (d).

The idea to keep. With a long chain of nested functions, find the value of every inner layer at the given point first. Then each chain-rule factor becomes a plain number, and you multiply them at the end.

Answer from the NTA final answer key. Solution by PrepWiser.