Question

nth derivative of y = sin^2x cos^3x is

(a) ^1⁄8 cos⁡(x + ^nπ⁄2) –^1⁄16 5^n cos⁡(x + ^nπ⁄2) – ^1⁄16 3^n cos⁡(3x + ^nπ⁄2)

(b) ^1⁄8 sin⁡(x+^nπ⁄2) –^1⁄16 5^n cos⁡(x + ^nπ⁄2) – ^1⁄16 3^n cos⁡(3x + ^nπ⁄2)

(c) ^1⁄8 cos⁡(x+^nπ⁄2) –^1⁄16 5^n sin⁡(x + ^nπ⁄2) – ^1⁄16 3^n sin⁡(3x + ^nπ⁄2)

(d) ^1⁄8 sin⁡(x + ^nπ⁄2) –^1⁄16 5^n sin⁡(x + ^nπ⁄2) – ^1⁄16 3^n sin⁡(3x + ^nπ⁄2)

I had been asked this question in homework.

My enquiry is from The nth Derivative of Some Elementary Functions in portion Differential Calculus of Engineering Mathematics

Answer 1

Right choice is (a) ^1⁄8 cos⁡(x + ^nπ⁄2) –^1⁄16 5^n cos⁡(x + ^nπ⁄2) – ^1⁄16 3^n cos⁡(3x + ^nπ⁄2)

To elaborate: y = sin^2x cos^2x cos(x)

y = ^1⁄4 sin^22x cos^2x cos(x)

y = ^1⁄8 (2sin^22x) cos(x)

y = ^1⁄8 (1 – cos4x) cos(x)

y = ^1⁄8 (1 – cos4x) cos(x)

y = ^1⁄8 cos(x) – ^1⁄8 cos4x cos(x)

y = ^1⁄8 cos(x) – ^1⁄16 (cos5x + cos(3x))

Now, nth derivative is

yn = ^1⁄8 cos⁡(x + ^nπ⁄2) – ^1⁄16 5^n cos⁡(x + ^nπ⁄2) – ^1⁄16 3^n cos⁡(3x + ^nπ⁄2).