Question

If
`x = \sqrt{a^{sin^(-1)t}}`,
`y =\sqrt{a^{cos^(-1)t}}`, show that
`(dy)/(dx)= -(y)/(x)`.

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Answer 1

`{ \qquad\qquad\huge\underline{{\sf Answer}}}`

Let's solve ~

`\qquad \sf  \dashrightarrow \: x = \sqrt{ {a}^{sin {}^( - 1)t } }`

here, let's differentiate it with respect to t ~

`\sf   \dashrightarrow \: (dx)/(dt) = \frac{1}{2 \sqrt{a {}^{sin {}^( - 1)t } } } * a {}^{sin {}^( - 1)t } \sdot ln(a) * \frac{1}{ \sqrt{1 - {x}^(2) } }`

`\sf  \dashrightarrow \: (dx)/(dt) = \frac{ \sqrt{ {a}^{sin {}^( - 1)t } } \sdot ln(a)}{2 \sqrt{1 - {x}^(2) } }`

`\sf \dashrightarrow \: \cfrac{dt}{dx} = \frac{2 \sqrt{1 - {x}^(2) } }{ \sqrt{a {}^{sin {}^( - 1) t} \sdot ln(a)} }`

Smililarly,

`\sf  \dashrightarrow \: (dy)/(dt) = \frac{1}{2 \sqrt{a {}^{cos{}^( - 1)t } } } * a {}^{cos {}^( - 1)t } \sdot ln(a) * \frac{ - 1}{ \sqrt{1 - {x}^(2) } }`

`\sf  \dashrightarrow \: (dy)/(dt) = - \frac{ \sqrt{ {a}^{cos {}^( - 1)t } } \sdot ln(a)}{2 \sqrt{1 - {x}^(2) } }`

Now : Lets get Required result ~

`\sf \dashrightarrow \: (dy)/(dx) = (dy )/(dt) * (dt)/(dx)`

`\sf \dashrightarrow \cfrac{dy}{dx} = - \frac{\sqrt{ {a}^{cos {}^( - 1)t } \sdot \cancel{ ln(a)}}}{ \cancel{2 \sqrt{1 - {x}^(2)}}} \sdot \frac{ \cancel{2 \sqrt{1 - {x}^(2)} } }{ \sqrt{a {}^{sin {}^( - 1) t} }\sdot \cancel{ln(a)}}`

`\sf \dashrightarrow \cfrac{dy}{dx} = - \frac{\sqrt{ {a}^{cos {}^( - 1)t } }}{ \sqrt{a {}^{sin {}^( - 1) t} }}`

`\sf \dashrightarrow \cfrac{dy}{dx} = - (y)/(x)`

[ since y =
`\sf{\sqrt{a^{cos^(-1)t}} }` and x =
`\sf{\sqrt{a^{sin^(-1)t}} }` ]

Answer 2

Explanation:

`\sf x = \sqrt{a^{sin^(-1) \ t}}\\\\\\Derivative \ rule:\boxed{(d(√(x)))/(dx)=(1)/(2)*x^{(-1)/(2)}=(1)/(2√(x))}`

`\sf \frac{d(\sqrt{a^{sin^(-1) \ t}}}{dt}=\frac{1}{2\sqrt{a^{sin^(-1) \ t}}}*\frac{d(a^{sin^(-1) \ t})}{dt}\\\\\\Derivative \ rule: \boxed{(d(a^(x)))/(dx)=log \ a *a^(x)}`

`\sf = \frac{1}{2\sqrt{a^{sin^(-1)} \ t}}*a^{sin^(-1) \ t}* log \ a *(d(Sin^(-1) \ t))/(dt)\\\\`

`Derivative \ rule:\boxed{(d(sin^(-1) \ x)/(dx)=(1)/(√(1-x^2))}`

`\sf = \frac{1}{2\sqrt{a^{sin^(-1) \ t}}}*a^{Sin^(-1) \ t}*log \ a*(1)/(√(1-x^2))}}}\\\\ = \frac{a^{Sin^(-1) \ t}*log \ a}{2\sqrt{a^{sin^(-1) \ t}}*√(1-x^2)}`

`\boxed{ \frac{a^{sin^(-1) \ t}}{\sqrt{a^{sin^(-1) \ t}}}=\frac{\sqrt{a^{sin^(-1) \ t}}*\sqrt{a^{sin^(-1) \ t}}}{\sqrt{a^{sin^(-1) \ t}}} = \sqrt{a^{sin^(-1) \ t}}}`

`\sf = \frac{a^{sin^(-1) \ t}*log \ a}{2√(1-x^2)}`

`\sf (dy)/(dt)=\frac{d(a^{cos^(-1) \ t})}{dt}`

`= \frac{1}{2\sqrt{a^{cos^(-1) \ t}}}*a^{cos^(-1) \ t}*log \ a *(-1)/(√(1-x^2))}\\\\\\=\frac{(-1)*a^{cos^(-1) \ t}*log \ a}{2*\sqrt{a^{cos^(-1) \ t}}*√(1-x^2)}`

`\sf = \frac{(-1)*\sqrt{a^{Cos^(-1) \ t}}* log \ a }{2√(1-x^2)}\\\\`

`\sf \bf (dy)/(dx)=(dy)/(dt) / (dx)/(dt)\\`

`\sf \bf = \frac{(-1)*\sqrt{a^{cos^(-1) \ t}}*log \ a}{2*√(1-x^2)} \ / \frac{\sqrt{a^{sin^(-1) \ t}} *log \ a}{2*√(1-x^2)}\\\\\\=\frac{(-1)*\sqrt{a^{cos^(-1) \ t}}*log \ a}{2*√(1-x^2)} \ * \frac{2*√(1-x^2)}{\sqrt{a^{sin^(-1) \ t}} *log \ a}\\\\= \frac{(-1)* \sqrt{a^{cos^(-1) \ t}} }{\sqrt{a^{sin^(-1) \ t}}}\\\\= (-y)/(x)`