Misc 27 - Find derivative: x2 cos pi/4 / sin x - Class 11

Misc 27 Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): x2 cos ﷮ π﷮4﷯﷯﷮ sin﷮x﷯﷯ Let f (x) = 𝑥2 cos ﷮ 𝜋﷮4﷯﷯﷮sin x﷯ Let u = x2 cos 𝜋﷮4﷯ & v = sin x So, f(x) = 𝑢﷮𝑣﷯ ∴ f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ Using quotient rule f’(x) = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ Finding u’ & v’ u = x2 cos 𝜋﷮4﷯ u’ = 2x2 –1 cos 𝜋﷮4﷯ = 2x cos 𝜋﷮4﷯ & v = sin x v’= cos x Now, f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ = 2𝑥 cos﷮ 𝜋﷮4﷯﷯ ( sin﷮𝑥)﷯ −( cos﷮𝑥) ( 𝑥﷮2﷯ cos﷮ 𝜋﷮4﷯) ﷯﷯﷮ 𝑠𝑖𝑛﷮2﷯ 𝑥﷯ = 2𝑥 𝑠𝑖𝑛 𝑥 cos ﷮ 𝜋﷮4﷯﷯ − 𝑥﷮2﷯ cos﷮𝑥 . cos ﷮ 𝜋﷮4﷯﷯﷯﷮ 𝑠𝑖𝑛﷮2﷯ 𝑥﷯ = 𝒙 𝐜𝐨𝐬 ﷮ 𝝅﷮𝟒﷯﷯ (𝟐 𝐬𝐢𝐧﷮𝒙 − 𝒙 𝐜𝐨𝐬﷮𝒙) ﷯﷯﷮ 𝒔𝒊𝒏﷮𝟐﷯ 𝒙﷯