Limits, Continuity & Differentiability
Differentiation
nta_abhyas_2025
Grade 12
Question:
If $x = \frac{1-\tan^2\theta}{1+\tan^2\theta}$ and $y = \frac{2\tan\theta}{1+\tan^2\theta}$, find $\frac{dy}{dx}$
Step-by-Step Solution
Key Concept: Recognize parametric equations in terms of double angles or use implicit differentiation on the constraint
Recognize that $x = \cos 2\theta$ and $y = \sin 2\theta$ using double angle formulas. Then $\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{2\cos 2\theta}{-2\sin 2\theta} = -\cot 2\theta$. Alternatively, from $x^2 + y^2 = 1$, differentiating implicitly: $2x + 2y\frac{dy}{dx} = 0$, so $\frac{dy}{dx} = -\frac{x}{y}$.
Correct Answer: 3