Find d y d x if, : y= (5 x^3-4 x^2-8 x )^9 - Vidyadip

Question

Find $\frac{d y}{d x}$ if, :
$
y=\left(5 x^3-4 x^2-8 x\right)^9
$

✓

Answer

Given : $y=\left(5 x^3-4 x^2-8 x\right)^9$
Let $u=5 x^3-4 x^2-8 x$
Then $y=u^9$
$
\begin{aligned}
\therefore \frac{d y}{d u} & =\frac{d}{d u}\left(u^9\right)=9 u^8 \\
& =9\left(5 x^3-4 x^2-8 x\right)^8
\end{aligned}
$
and $\frac{d u}{d x}=\frac{d}{d x}\left(5 x^3-4 x^2-8 x\right)$
$
\begin{aligned}
& =5 \frac{d}{d x}\left(x^3\right)-4 \frac{d}{d x}\left(x^2\right)-8 \frac{d}{d x}(x) \\
& =5 \times 3 x^2-4 \times 2 x-8 \times 1 \\
& =15 x^2-8 x-8
\end{aligned}
$
$
\begin{aligned}
\therefore \frac{d y}{d x} & =\frac{d y}{d u} \cdot \frac{d u}{d x}\\
& =9\left(5 x^3-4 x^2-8 x\right)^8\left(15 x^2-8 x-8\right) .
\end{aligned}
$

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