If $y=(tan~x)^{x}$, then find $\frac{dy}{dx}$ .

If $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y),$ prove that $\frac{dy}{dx}=\sqrt{\frac{1-y^{2}}{1-x^{2}}} .$

The derivative of \(2^{x}\) w.r.t. \(3^{x}\) is:

Derivative of \(e^{2x}\) with respect to \(e^{x}\), is:

Amusement parks heavily rely on mathematical curves to design roller coaster tracks, balancing intense thrill with absolute safety. For a safe ride, the track must have no breaks (it must be continuous) and no sharp, sudden corners (it must be differentiable/smooth). An engineering team is designing a transition segment of a new coaster. The height $h(x)$ (in meters) of the track at a horizontal distance $x$ (in meters) from the starting platform is modeled by the following piecewise function: $$h(x) = \begin{cases} \frac{1}{2}x^2 + 2x, & 0 \leq x < 2 \\ ax + b, & 2 \leq x \leq 5 \end{cases}$$ Based on the given information, answer the following questions: (i) For the track to have no broken rails at the transition point $x = 2$, it must be continuous. Formulate the mathematical equation relating $a$ and $b$ to ensure this continuity. (1 Mark) (ii) Find the left-hand derivative (the slope of the track just before the transition point) at $x = 2$. (1 Mark) (iii) To ensure the transition at $x = 2$ is perfectly smooth (differentiable), calculate the exact values of $a$ and $b$ that the engineers must use. (2 Marks) — OR — (iii) If a junior engineer mistakenly designed the second track segment using $a = 3$ and $b = 0$, verify whether the track is mathematically safe (smooth and differentiable) at $x = 2$. Justify your answer. (2 Marks)