Find d y d x if x = e ^ 3 t , y =e^ t . - Vidyadip

Question

Find $\frac{d y}{d x}$ if $x = e ^{3 t }, y =e^{\sqrt{t}}$.

✓

Answer

$
x = e ^{3 t }, y =e^{\sqrt{t}}
$
Differentiating $x$ and $y$ w.r.t. $t$, we get
$
\begin{aligned}
\frac{d x}{d t} & =\frac{d}{d t}\left(e^{3 t}\right)=e^{3 t} \cdot \frac{d}{d t}(3 t) \\
& =e^{3 t} \times 3=3 e^{3 t}
\end{aligned}
$
$
\text { and } \frac{d y}{d t}=\frac{d}{d t}(e \sqrt{t})=e \sqrt{t} \cdot \frac{d}{d t}(\sqrt{t})
$
$
\begin{aligned}
& =e \sqrt{t} \times \frac{1}{2 \sqrt{t}}=\frac{e \sqrt{t}}{2 \sqrt{t}} \\
\therefore \frac{d y}{d x} & =\frac{(d y / d t)}{(d x / d t)}=\frac{\left(\frac{e \sqrt{t}}{2 \sqrt{t}}\right)}{3 e^{3 t}} \\
& =\frac{1}{6 \sqrt{t}} \cdot e^{(\sqrt{t}-3 t)} .
\end{aligned}
$

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