1. Problem: Find the derivatives and analyze curves based on given problems. 2. (3a) Let $y = \tan^{-1}\left(\frac{4\sqrt{x}}{1 - 4x}\right)$. Use the chain rule and derivative of arctan:\ $$\frac{dy}{dx} = \frac{1}{1 + \left(\frac{4\sqrt{x}}{1-4x}\right)^2} \cdot \frac{d}{dx}\left(\frac{4\sqrt{x}}{1 - 4x}\right)$$ Calculate numerator and denominator derivatives carefully using quotient and chain rule. 3. (3b) Let $y = \tanh^{-1}\left(\frac{2x}{1+x^2}\right)$. Recall $$\frac{d}{dx}\tanh^{-1}(u) = \frac{1}{1-u^2}\frac{du}{dx}$$ Here, $$u = \frac{2x}{1+x^2}$$ Calculate $du/dx$ using quotient rule and simplify. Substitute back to find $dy/dx$. 4. (4a) Given $y = (x - 2)^2 (x - 7)$, expand to $y = (x^2 - 4x + 4)(x - 7) = x^3 -11x^2 + 32x - 28$. Find second derivative $y''$ for points of inflexion by solving $y''=0$. 5. (4b) For $y = 4x^3 + 3x^2 -18x - 9$, compute $y'' = 24x + 6$. Set $y''=0$ to find inflection points. 6. (5) For the function $y(3x - 2) = (3x -1)^2$, define $t = 3x - 2$, then $y(t) = (t + 1)^2$. Find $dy/dt$, calculate where derivative is zero for max/min, use chain rule reprising in $x$, sketch graph. 7. (6) Given $y=12 \ln x + x^2 - 10x$, find $dy/dx = 12/x + 2x - 10$, set to zero to find critical points. Find $d^2y/dx^2$ to classify max, min, or inflection. 8. (7) For implicit $4x^2 + 8xy + 9y^2 - 8x - 24y + 4 = 0$, differentiate implicitly to find $dy/dx$. Show that when $dy/dx=0$, $x + y = 1$. Then differentiate again for $d^2y/dx^2$ and show the given formula. Find max and min of $y$ by analyzing derivative conditions. 9. (14) For power function $P = T v - \frac{\omega v^3}{g}$, differentiate wrt $v$, set derivative to zero: $$\frac{dP}{dv} = T - \frac{3\omega v^2}{g} = 0 \Rightarrow v = \sqrt{\frac{Tg}{3\omega}}$$ This $v$ gives max transmitted power. 10. (15) For a cone with curved surface area $A$, express volume $V$ and optimize. Shows maximum volume ratio of height to base radius is $\sqrt{2}:1$ by solving derivative condition. Final answers are described in the explanations above. Each problem involves standard calculus and algebra techniques.