Question

If \( x = \sin t - \cos t \) and \( y = \sin t \cos t \), find the value of \(\frac{dy}{dx}\) evaluated at \( t = \frac{\pi}{4} \).

Hint:

Always simplify using trigonometric double angle formulas before differentiating. It saves huge amounts of time compared to doing product rules. For example, rewriting \( y = \frac{1}{2}\sin 2t \) directly gave us \(\cos 2t\) as the derivative in a single line!

Answer 1

Concept: This problem involves parametric differentiation. When variables \(x\) and \(y\) are given as functions of an independent parameter \(t\), the derivative of \(y\) with respect to \(x\) is computed using the formula: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \] provided that \( \frac{dx}{dt} \neq 0 \). If the denominator is zero while the numerator is non-zero, the derivative becomes undefined (tending to infinity).

Step 1: Differentiate \( x \) with respect to parameter \( t \).
The parametric equation for \( x \) is: \[ x = \sin t - \cos t \] Differentiating with respect to \( t \): \[ \frac{dx}{dt} = \frac{d}{dt}(\sin t) - \frac{d}{dt}(\cos t) \] Using standard derivative formulas, \( \frac{d}{dt}(\sin t) = \cos t \) and \( \frac{d}{dt}(\cos t) = -\sin t \): \[ \frac{dx}{dt} = \cos t - (-\sin t) = \cos t + \sin t \quad \cdots (1) \]

Step 2: Differentiate \( y \) with respect to parameter \( t \).
The parametric equation for \( y \) is given by: \[ y = \sin t \cos t \] We can use the product rule or simplify using the double-angle identity: \( \sin(2t) = 2\sin t \cos t \Rightarrow y = \frac{1}{2}\sin(2t) \). Let us differentiate this simplified form: \[ \frac{dy}{dt} = \frac{d}{dt}\left(\frac{1}{2}\sin(2t)\right) = \frac{1}{2} \cdot \cos(2t) \cdot \frac{d}{dt}(2t) \] \[ \frac{dy}{dt} = \frac{1}{2} \cdot \cos(2t) \cdot 2 = \cos(2t) \quad \cdots (2) \]

Step 3: Combine derivatives to form the expression for \(\frac{dy}{dx}\).
Using the parametric derivative formula: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{\cos(2t)}{\cos t + \sin t} \]

Step 4: Evaluate the derivative at the point \( t = \frac{\pi}{4} \).
Substitute \( t = \frac{\pi}{4} \) into the expressions:
• Numerator evaluation: \[ \cos\left(2 \cdot \frac{\pi}{4}\right) = \cos\left(\frac{\pi}{2}\right) = 0 \]
• Denominator evaluation: \[ \cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \] Now, substitute these computed values back into the expression for \(\frac{dy}{dx}\): \[ \left.\frac{dy}{dx}\right|_{t=\frac{\pi}{4}} = \frac{0}{\sqrt{2}} = 0 \] Let us re-verify carefully: The previous step stated \(C\) as correct due to potential alternative reading but the clean math shows \(0 / \sqrt{2} = 0\). Let us fix the option matching. Option (A) is the true correct calculation answer. Let us rewrite correct answer line as (A) 0. % Correct Answer Updated Correct Answer: (A) \(\frac{dy}{dx} = 0\)