Find d y d x if : x=t-t, y=t+t - Vidyadip

Question

Find $\frac{d y}{d x}$ if : $x=t-\sqrt{t}, y=t+\sqrt{t}$

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Answer

Given, $y=t+\sqrt{t}$
Differentiate $w$. r.t.t
$\frac{d y}{d t}=\frac{d}{d t}(t+\sqrt{t})=1+\frac{1}{2 \sqrt{t}}$
$\frac{d y}{d t}=\frac{2 \sqrt{t}+1}{2 \sqrt{t}}$
And, $x=t-\sqrt{t}$
Differentiate $w$. r. t. $t$
$\frac{d x}{d t}=\frac{d}{d t}(t-\sqrt{t})=1-\frac{1}{2 \sqrt{t}}$
$\frac{d x}{d t}=\frac{2 \sqrt{t}-1}{2 \sqrt{t}}$
Now, $\frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{\frac{2 \sqrt{t}+1}{2 \sqrt{t}}}{\frac{2 \sqrt{t}-1}{2 \sqrt{t}}} \ldots[$ From (I) and (II) $]$
$
\therefore \quad \frac{d y}{d x}=\frac{2 \sqrt{t}+1}{2 \sqrt{t}-1}
$

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