Find d y d x if, : x=e^ 3 t , y=e^ (4 t+5)

Question

Find $\frac{d y}{d x}$ if, :
$
x=e^{3 t}, y=e^{(4 t+5)}
$

✓

Answer

$
x=e^{3 t}, y=e^{(4 t+5)}
$
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 \times 1=3 e^{3 t}
\end{aligned}
$
$
\text { and } \begin{aligned}
\frac{d y}{d t} & =\frac{d}{d t}\left[e^{(4 t+5)}\right]=e^{(4 t+5)} \cdot \frac{d}{d t}(4 t+5) \\
& =e^{(4 t+5)} \times(4 \times 1+0)=4 e ^{(4 t+5)}
\end{aligned}
$
$
\begin{aligned}
\therefore \frac{d y}{d x} & =\frac{(d y / d t)}{(d x / d t)}=\frac{4 e^{(4 t+5)}}{3 e^{3 t}} \\
& =\frac{4}{3} e^{4 t+5-3 t}=\frac{4}{3} e^{t+5}
\end{aligned}
$

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