Question

If \( x = e^{\sin^{-1} t}, y = e^{\cos^{-1} t} \), find \( \frac{dy}{dx} \) at \( t = \frac{1}{\sqrt{2}} \).

Hint:

Alternative Shortcut: Notice \( xy = e^{\sin^{-1} t} \cdot e^{\cos^{-1} t} = e^{\sin^{-1} t + \cos^{-1} t} = e^{\pi/2} \).
Differentiating \( xy = e^{\pi/2} \) implicitly gives \( x \frac{dy}{dx} + y = 0 \), so \( \frac{dy}{dx} = -y/x \). This is much faster!

Answer 1

Concept:
• Derivatives of parametric functions: \( \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \).
• Chain Rule: \( \frac{d}{dx}[e^{f(x)}] = e^{f(x)} \cdot f'(x) \).
• Standard Derivatives: \( \frac{d}{dt}(\sin^{-1} t) = \frac{1}{\sqrt{1-t^2}} \) and \( \frac{d}{dt}(\cos^{-1} t) = \frac{-1}{\sqrt{1-t^2}} \).
• Trigonometric Identity: \( \sin^{-1} t + \cos^{-1} t = \frac{\pi}{2} \).

Step 1: Calculate the derivative of \( x \) with respect to \( t \)
Given \( x = e^{\sin^{-1} t} \).
Differentiating with respect to \( t \) using the chain rule:
\[ \frac{dx}{dt} = e^{\sin^{-1} t} \cdot \frac{d}{dt}(\sin^{-1} t) \]
\[ \frac{dx}{dt} = e^{\sin^{-1} t} \cdot \frac{1}{\sqrt{1-t^2}} = \frac{x}{\sqrt{1-t^2}} \]

Step 2: Calculate the derivative of \( y \) with respect to \( t \)
Given \( y = e^{\cos^{-1} t} \).
Differentiating with respect to \( t \) using the chain rule:
\[ \frac{dy}{dt} = e^{\cos^{-1} t} \cdot \frac{d}{dt}(\cos^{-1} t) \]
\[ \frac{dy}{dt} = e^{\cos^{-1} t} \cdot \left( \frac{-1}{\sqrt{1-t^2}} \right) = \frac{-y}{\sqrt{1-t^2}} \]

Step 3: Find the expression for \( \frac{dy}{dx} \)
Using the parametric derivative formula:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{\frac{-y}{\sqrt{1-t^2}}}{\frac{x}{\sqrt{1-t^2}}} \]
\[ \frac{dy}{dx} = -\frac{y}{x} \]

Step 4: Evaluate at the given point \( t = 1/\sqrt{2} \)
At \( t = \frac{1}{\sqrt{2}} \):
\[ \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4} \implies x = e^{\pi/4} \]
\[ \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4} \implies y = e^{\pi/4} \]
Substituting these into the derivative expression:
\[ \left( \frac{dy}{dx} \right)_{t = 1/\sqrt{2}} = -\frac{e^{\pi/4}}{e^{\pi/4}} = -1 \]