If y=(√x+1)(x²-√x)/(x √x+x+√x)+1/15(3 cos ² x-5) cos ³ x, then 96 y′(π/6) is equal to :

Solution

Maths sir · 4 steps

Do not differentiate that monster fraction: put t=√x and it collapses to x-1.

1. Tame the fraction. Put t = √x. The numerator is (t+1)(t⁴ - t) = (t+1) t (t-1)(t²+t+1) and the denominator is t³+t²+t = t (t²+t+1). Cancel t(t²+t+1) to leave (t+1)(t-1) = t² - 1 = x - 1, whose derivative is 1.
2. Expand the cosine part. 1/15(3 cos² x - 5)cos³ x = 1/5 cos⁵ x - 1/3 cos³ x. By the chain rule, d/dx cosⁿ x = -n cosⁿ⁻¹ x sin x.

   y' = 1 - cos⁴ x sin x + cos² x sin x = 1 + cos² x sin x (1-cos² x) = 1 + sin³ x cos² x

3. Substitute the angle. sin π/6 = 1/2, so sin³ x = 1/8, and cos² π/6 = 3/4. Then y'(π/6) = 1 + 3/32 = 35/32.
4. So. 96 × 35/32 = 3 × 35 = 105.

The common slip: 87. Dropping the minus sign in d/dx cosⁿ x = -n cosⁿ⁻¹ x sin x gives y' = 1 - 3/32 = 29/32, so the answer becomes 87, not 105.

The idea to keep. When an expression looks ugly, simplify before you differentiate. A substitution like t=√x often makes factors cancel, and powers of cos x usually combine into a neat product.

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