Find the nth derivative of x²/(x-a)(x-b)

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Learn how to find the nth derivative of x²/(x-a)(x-b) using partial fractions and differentiation techniques. Follow our detailed step-by-step solution to master this calculus problem

Find the nth derivative of x²/(x-a)(x-b)

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

Solution:- Let,  y = \dfrac{x^2}{(x-a)(x-b)}

Using partial fraction

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

Taking LCM we get

\dfrac{x^2}{(x-a)(x-b)} = \dfrac{(x-a)(x-b) +A(x-b)+B(x-a)}{(x-a)(x-b)}

⇒ x^2 = (x-a)(x-b) +A(x-b)+B(x-a)

Let, x = a

(a)² =0 + A(a-b) + 0

⇒ A = a²/(a-b)

Let, x = b

(b)² = 0 + 0 +B(b-a)

⇒ B = b²/(b-a)

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

\dfrac{x^2}{(x-a)(x-b)} = 1 + \dfrac{a^2/(a-b)}{(x - a)} + \dfrac{b^2/(b-a)}{(x-b)}

⇒ y = 1 + \dfrac{a^2/(a-b)}{(x - a)} + \dfrac{b^2/(b-a)}{(x-b)}

⇒ y = 1 + \dfrac{a^2}{(a-b)} (x-a)^{-1} + \dfrac{b^2}{(b-a)}(x - b)^{-1}

Now, differentiate this equation with respect to x

y_1 = 0 + \dfrac{a^2}{(a-b)}(-1)(x-a)^{-2} + \dfrac{b^2}{(b-a)}(-1)(x-b)^{-2}

Again differentiate with respect to x

y_2 = \dfrac{a^2}{(a-b)}(-1)(-2)(x-a)^{-3} + \dfrac{b^2}{(b-a)}(-1)(-2)(x-b)^{-3}

⇒ y_2 = \dfrac{a^2}{(a-b)}(-1)^2(2)!(x-a)^{-3} + \dfrac{b^2}{(b-a)}(-1)^2(2)!(x-b)^{-3}

Now, differentiate this equation (n-2) times

y_n = \dfrac{a^2}{(a-b)}(-1)^n(n)!(x-a)^{-n-1} + \dfrac{b^2}{(b-a)}(-1)^n(n)!(x-b)^{-n-1}}

⇒ y_n = \dfrac{a^2}{(a-b)}\dfrac{(-1)^n n!}{(x-a)^{n+1}} + \dfrac{b^2}{(b-a)}\dfrac{(-1)^n n!}{(x-b)^{n+1}}

⇒ y_n = \dfrac{(-1)^n(n)!}{(a-b)}\left[\dfrac{a^2}{(x-a)^{n+1}} - \dfrac{b^2}{(x-b)^{n+1}} \right]

Some other solution

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