if y x²upon (x-1)² into (x+2) find the nth derivative of y

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See full solution of  how to find the nth derivative of the function x²upon (x-1)² into (x+2) using partial fractions and differentiation techniques. Step-by-step solution provided to simplify and solve this complex calculus problem

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

SOlution: Given, y = \dfrac{x^2}{(x-1)^2(x+2)}

Using partial fraction

\dfrac{x^2}{(x-1)^2(x+2)} = \dfrac{A}{(x-1)} + \dfrac{B}{(x-1)^2} + \dfrac{C}{(x+2)} .  .  . (i)

⇒ \dfrac{x^2}{(x-1)^2(x+2)} = \dfrac{A(x-1)(x+2)+B(x+2) + C(x-1)^2}{(x-1)^2(x+2)}

⇒ x² = A(x-1)(x+2)+B(x+2) + C(x-1)²

Let x = 1

1 = 0 + B(1 + 2) + C(0)

⇒ B = 1/3

Let x = -2

(-2)² = 0 + 0 + (-2-1)²

⇒ 4 = 9C

⇒  C = 4/9

Let x = 0

0 = A(-1)(2) + B(2) + C(1)

⇒  0 = -2A + 2(1/3) + 4/9

⇒ -2/3 – 4/9 = -2A

⇒ \dfrac{-6 - 4}{9} = -2A

⇒ -2A =  -10/9

⇒ A =    5/9

Replacing the value of A,B and C in equation (i)

we get,

\dfrac{x^2}{(x-1)^2(x+2)} = \dfrac{5/9}{(x-1)} + \dfrac{1/3}{(x-1)^2} + \dfrac{4/9}{(x+2)}

⇒ y = \frac{5}{9}(x-1)^{-1} + \frac{1}{3}(x - 1)^{-2} + \frac{4}{9}(x + 2)^{-1}

Now differentiate with respect to x

y_1 = \dfrac{d}{dx}y = \frac{5}{9}(-1)(x -1 )^{-2} + \frac{1}{3}(-2)(x-1)^{-3} + \frac{4}{9}(-1)(x + 2)^{-2}

Again diffrentiate with respect to x

y_2 = \frac{5}{9}(-1)(-2)(x -1 )^{-3} + \frac{1}{3}(-2)(-3)(x-1)^{-4} + \frac{4}{9}(-1)(-2)(x + 2)^{-3}

⇒ y_2 = \frac{5}{9}(-1)^2(2)!(x -1 )^{-3} + \frac{1}{3}(-1)^2(3)!(x-1)^{-4} + \frac{4}{9}(-1)^2(2)!(x + 2)^{-3}

Now differentiate (n-2) times this equation with respect to x

We get,

y_n = \frac{5}{9}(-1)^n (n)! (x - 1)^{-n-1} + \frac{1}{3}(-1)^n (n+1)! (x-1)^{-n-2} + \frac{4}{9}(-1)^n (n)! (x + 2)^{-n-1}

⇒ y_n = \dfrac{5(-1)^n n!}{9(x - 1)^{n+1}} + \dfrac{(-1)^n (n+1)!}{3(x-1)^{n+2}} + \dfrac{4(-1)^n n!}{9(x+ 2)^{n+1}}

Some other question

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

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

Solution:- See full solution

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