Find the derivative of the following functions
Question:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): $\frac{1}{a x^{2}+b x+c}$

Solution:

Let $f(x)=\frac{1}{a x^{2}+b x+c}$

By quotient rule,

$f^{\prime}(x)=\frac{\left(a x^{2}+b x+c\right) \frac{d}{d x}(1)-\frac{d}{d x}\left(a x^{2}+b x+c\right)}{\left(a x^{2}+b x+c\right)^{2}}$

$=\frac{\left(a x^{2}+b x+c\right)(0)-(2 a x+b)}{\left(a x^{2}+b x+c\right)^{2}}$

$=\frac{-(2 a x+b)}{\left(a x^{2}+b x+c\right)^{2}}$

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