Question

how to find nth derivative of x^2/(x-a)(x-b)(x-c).
tried out splitting up into partial fraction and ended up with the result
a^2=A(a-b)(a-c)
b^2=B(b-a)(b-c)
c^2=C(c-a)(c-b)

Answer 1

\[\begin{align*}f(x)=\frac{x^2}{(x-a)(x-b)(x-c)}&=\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c}\\\\
x^2&=A(x-b)(x-c)+B(x-a)(x-c)+C(x-a)(x-b)
\end{align*}\]
Using your method, you have
\[\begin{array}{c|c}
x=\cdots&\text{equation}&\text{coefficient}\\
\hline
a&a^2=A(a-b)(a-c)&A=\dfrac{a^2}{(a-b)(a-c)}\\\\
b&b^2=B(b-a)(b-c)&B=\dfrac{b^2}{(b-a)(b-c)}\\\\
c&c^2=C(c-a)(c-b)&C=\dfrac{c^2}{(c-a)(c-b)}
\end{array}\]
So,
\[f(x)=\frac{a^2}{(x-a)(a-b)(a-c)}+\frac{b^2}{(x-b)(b-a)(b-c)}+\frac{c^2}{(x-c)(c-a)(c-b)}\]

Answer 2

thank you!!

Answer 3

yw