<s>[INST] Ugly Differentiation If $$y=\frac{ax^2}{(x-a)(x-b)(x-c)} + \frac{bx}{(x-b)(x-c)}+\frac{c}{x-c}+1$$
Then prove that $$\frac{dy}{dx}=\frac{y}{x}(\frac{a}{a-x}+\frac{b}{b-x}+\frac{c}{c-x})$$
What is the smartest way to solve this?
 [/INST] HINT:
Use Partial Fraction Decomposition,
$$\frac{ax^2}{(x-a)(x-b)(x-c)} + \frac{bx}{(x-b)(x-c)}+\frac{c}{c-x}=\dfrac A{x-a}+\dfrac B{x-b}+\dfrac C{x-c}$$
$$\implies ax^2+bx(x-a)+c(x-a)(x-b)=A(x-b)(x-c)+B(x-c)(x-a)+C(x-a)(x-b)$$
Compare the constants and the coefficients of $x,x^2$ to find $A,B,C$
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