Problem 17: Find $\frac{dr}{d\theta}$ if $\sqrt{\theta} + \sqrt{r} = 1$. 1. Rewrite equation: $\theta^{1/2} + r^{1/2} = 1$ 2. Differentiate both sides with respect to $\theta$ implicitly: $$\frac{d}{d\theta}(\theta^{1/2}) + \frac{d}{d\theta}(r^{1/2}) = \frac{d}{d\theta}(1)$$ 3. Compute derivatives: $$\frac{1}{2} \theta^{-1/2} + \frac{1}{2} r^{-1/2} \frac{dr}{d\theta} = 0$$ 4. Solve for $\frac{dr}{d\theta}$: $$\frac{1}{2} r^{-1/2} \frac{dr}{d\theta} = -\frac{1}{2} \theta^{-1/2}$$ $$\frac{dr}{d\theta} = -\frac{\theta^{-1/2}}{r^{-1/2}} = -\frac{\sqrt{r}}{\sqrt{\theta}}$$ Problem 18: Find $\frac{dr}{d\theta}$ if $r - 2\sqrt{\theta} = \frac{3}{2} \theta^{2/3} + \frac{4}{3} \theta^{3/4}$. 1. Given: $$r - 2 \theta^{1/2} = \frac{3}{2} \theta^{2/3} + \frac{4}{3} \theta^{3/4}$$ 2. Differentiate both sides w.r.t $\theta$: $$\frac{dr}{d\theta} - 2 \cdot \frac{1}{2} \theta^{-1/2} = \frac{3}{2} \cdot \frac{2}{3} \theta^{-1/3} + \frac{4}{3} \cdot \frac{3}{4} \theta^{-1/4}$$ 3. Simplify derivatives: $$\frac{dr}{d\theta} - \theta^{-1/2} = \theta^{-1/3} + \theta^{-1/4}$$ 4. Solve for $\frac{dr}{d\theta}$: $$\frac{dr}{d\theta} = \theta^{-1/2} + \theta^{-1/3} + \theta^{-1/4}$$ Problem 19: Find $\frac{dr}{d\theta}$ if $\sin(r\theta) = \frac{1}{2}$. 1. Differentiate both sides w.r.t $\theta$: $$\cos(r\theta) \cdot \frac{d}{d\theta}(r\theta) = 0$$ 2. Use product rule on $r\theta$: $$\cos(r\theta) (r + \theta \frac{dr}{d\theta}) = 0$$ 3. Since $\cos(r\theta) \neq 0$ (otherwise no variation), set inside bracket to zero: $$r + \theta \frac{dr}{d\theta} = 0$$ 4. Solve for $\frac{dr}{d\theta}$: $$\frac{dr}{d\theta} = -\frac{r}{\theta}$$ Problem 20: Find $\frac{dr}{d\theta}$ given $\cos r + \cot \theta = e^{r\theta}$. 1. Differentiate both sides w.r.t $\theta$: $$-\sin r \frac{dr}{d\theta} - \csc^2 \theta = e^{r\theta} (r + \theta \frac{dr}{d\theta})$$ 2. Group terms with $\frac{dr}{d\theta}$: $$-\sin r \frac{dr}{d\theta} - e^{r\theta} \theta \frac{dr}{d\theta} = e^{r\theta} r + \csc^2 \theta$$ 3. Factor out $\frac{dr}{d\theta}$: $$\frac{dr}{d\theta} (-\sin r - e^{r\theta} \theta) = e^{r\theta} r + \csc^2 \theta$$ 4. Solve for $\frac{dr}{d\theta}$: $$\frac{dr}{d\theta} = \frac{e^{r\theta} r + \csc^2 \theta}{-\sin r - e^{r\theta} \theta} = -\frac{e^{r\theta} r + \csc^2 \theta}{\sin r + e^{r\theta} \theta}$$