List of top Mathematics Questions on Continuity and differentiability asked in PSEB XII

If \( u = x^y, v = y^x \) and quantity \( y \) remains 3 times the quantity \( x \) then find that amongst quantities \( u \) and \( v \), which changes more rapidly with respect to quantity \( x \) when \( x = 1 \). (Take \( \log_e 3 = 1.09 \))
If \( x = \frac{1-t^2}{1+t^2} , y = \frac{2t}{1+t^2} \) then prove that \( \frac{dy}{dx} + \frac{x}{y} = 0 \).
Correct Answer: Proved
Find \( \frac{d}{dx}(x^x) \) and evaluate whether this result is true \( \forall x \in \mathbb{R} \).