Find d y d x if, : y=e^ 5 x^2-2 x+4 - Vidyadip

Question

Find $\frac{d y}{d x}$ if, :
$
y=e^{5 x^2-2 x+4}
$

✓

Answer

Given : $y=e^{5 x^2-2 x+4}$
Let $u=5 x^2-2 x+4$
Then $y=e^u$
$
\begin{aligned}
\therefore \frac{d y}{d u} & =\frac{d}{d u}\left(e^u\right)=e^u \\
& =e^{5 x^2-2 x+4}
\end{aligned}
$
and $\frac{d u}{d x}=\frac{d}{d x}\left(5 x^2-2 x+4\right)$
$
\begin{aligned}
& =5 \frac{d}{d x}\left(x^2\right)-2 \frac{d}{d x}(x)+\frac{d}{d x}(4) \\
& =5 \times 2 x-2 \times 1+0=10 x-2 \\
\therefore \frac{d y}{d x} & =\frac{d y}{d u}-\frac{d u}{d x} \\
& =e^{5 x^2-2 x+4} \times(10 x-2) \\
& =(10 x-2) e^{5 x^2-2 x+4} .
\end{aligned}
$

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