Find the value of nth derivative of y=cos x cos 2x cos 3x

31/05/2024 by Team Gmath

Learn how to Find the value of nth derivative of y=cos x cos 2x cos 3x using partial fractions and differentiation techniques. Follow our detailed step-by-step solution to master this calculus problem

Q 10:- Find the value of nth derivative of $y=\cos x\cos 2x \cos 3x$ .

Solution:- Let $y = \cos x\cos 2x \cos 3x$

⇒ $y =\frac{1}{2}[2\cos 3x\cos x]\cos 2x$

⇒ $y =\frac{1}{2}[\cos 4x + \cos 2x]\cos 2x$

⇒ $y =\frac{1}{4}[2\cos 4x\cos 2x + 2\cos 2x\cos 2x]$

⇒ $y =\frac{1}{4}[2\cos 4x\cos 2x + 2\cos^2 2x]$

⇒ $y =\frac{1}{4}[\cos 6x + \cos 2x + \cos 4x + 1]$

⇒ $y =\frac{1}{4}[\cos 6x + \cos 4x + \cos 2x + 1]$

Differentiate with respect to x

$y_1 =\frac{1}{4}[-6\sin 6x - 4\sin 4x - 2\sin 2x +0]$

⇒ $y_1 =\frac{1}{4}[6\cos (6x + \pi/2) + 4\cos (4x + \pi/2) + 2\cos (2x + \pi/2)]$ . . . (i)

Again diffrentiate with respect to x of equation (i)

$y_2 =\frac{1}{4}[6^2\cos (6x + 2\pi/2) + 4^2\cos (4x + 2\pi/2) + 2^2\cos (2x + 2\pi/2)]$

Now differentiate this equation with respect to x for $y_n$

$y_2 =\frac{1}{4}[6^n\cos (6x + n\pi/2) + 4^n\cos (4x + n\pi/2) + 2^n\cos (2x + n\pi/2)]$

Some other question

Q 1: If $y = \dfrac{x^2}{(x-1)^2(x+2)}$ , find nth derivative of y.

Solution: See full solution

Q 2: Find the nth derivative of $\dfrac{x^2}{(x-a)(x-b)}$ .

Solution:- See full solution

Q 3:- Find the nth derivative of $\tan^{-1}\left[\dfrac{1+x}{1-x}\right]$

Solution:- See full solution

Q 4:- If y = sin³ x find nth term of y.

Solution:- See full solution

Q 5:- Find nth derivative of $\tan^{-1}(x/a)$ .

Solution:- See full solution

Q 6:- Find nth derivative of $e^x.x$ .

Solution:- See full solution

Q 7:- Find nth derivative of the function $\dfrac{1-x}{1+x}$ .

Solution:- See full solution

Q 8:- Find nth derivative of $\lox x^2$

Solution:- Find full solution

Q 9:- Find nth derivative of the funcrtion $e^x.\sin^2 x$ .

Solution:- See full solution

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