Find d y d x if, : y= (a x^2+b x+c ) - Vidyadip

Question

Find $\frac{d y}{d x}$ if, :
$
y=\log \left(a x^2+b x+c\right)
$

✓

Answer

$
\text { Given } y=\log \left(a x^2+b x+c\right)
$
Let $u = ax x ^2+ bx + c$
Then $y=\log u$
$
\begin{aligned}
\therefore \frac{d y}{d u} & =\frac{d}{d u}(\log u)=\frac{1}{u} \\
& =\frac{1}{a x^2+b x+c}
\end{aligned}
$
and $\frac{d u}{d x}=\frac{d}{d x}(a x+b x+c)$
$
\begin{aligned}
& =a \frac{d}{d x}\left(x^2\right)+b \frac{d}{d x}(x)+\frac{d}{d x}(c) \\
& =a \times 2 x+b \times 1 \times 0=2 a x+b \\
\therefore \frac{d y}{d x} & =\frac{d y}{d u} \cdot \frac{d u}{d x}=\frac{1}{a x^2+b x+c} \times(2 a x+b) \\
& =\frac{2 a x+b}{a x^2+b x+c} .
\end{aligned}
$

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