1. Problem: Find $\frac{dy}{dx}$ if $f(x,y) = c$ (constant). Formula: For implicit functions, $\frac{dy}{dx} = -\frac{f_x}{f_y}$ where $f_x = \frac{\partial f}{\partial x}$ and $f_y = \frac{\partial f}{\partial y}$. Explanation: Since $f(x,y)$ is constant, total derivative $df = f_x dx + f_y dy = 0$. Rearranging gives $\frac{dy}{dx} = -\frac{f_x}{f_y}$. Answer: (b) $-\frac{f_x}{f_y}$. 2. Problem: Find $\frac{\partial f}{\partial x}$ if $f(x,y) = x^y$. Formula: Treating $y$ as constant, $\frac{\partial}{\partial x} x^y = y x^{y-1}$. Answer: (a) $y x^{y-1}$. 3. Problem: Find degree of homogeneous function $f(x,y) = \log \left(\frac{x}{y}\right)$. Explanation: A function $f$ is homogeneous of degree $n$ if $f(tx, ty) = t^n f(x,y)$. Here, $f(tx, ty) = \log \left(\frac{tx}{ty}\right) = \log \left(\frac{x}{y}\right) = f(x,y)$, so degree $=0$. Answer: (a) 0. 4. Problem: Find $\frac{\partial(u,v)}{\partial(x,y)}$ where $u = x + y$, $v = x - y$. Formula: Jacobian determinant $= \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix}$. Calculate: $\frac{\partial u}{\partial x} = 1$, $\frac{\partial u}{\partial y} = 1$, $\frac{\partial v}{\partial x} = 1$, $\frac{\partial v}{\partial y} = -1$. Jacobian $= (1)(-1) - (1)(1) = -1 - 1 = -2$. Answer: (c) -2. 5. Problem: Find degree of homogeneous function $f(x,y) = \frac{x^2 + y^2}{xy}$. Rewrite: $f(x,y) = \frac{x^2}{xy} + \frac{y^2}{xy} = \frac{x}{y} + \frac{y}{x}$. Check homogeneity: $f(tx, ty) = \frac{tx}{ty} + \frac{ty}{tx} = \frac{x}{y} + \frac{y}{x} = f(x,y)$, so degree $=0$. Answer: (a) 0. 6. Problem: Find equation of tangent to $y^2 = 4ax$ at point $(0,0)$. Differentiate implicitly: $2y \frac{dy}{dx} = 4a$, so $\frac{dy}{dx} = \frac{4a}{2y} = \frac{2a}{y}$. At $(0,0)$, slope is undefined (vertical tangent). Equation of tangent line at $(0,0)$ is $x=0$. 7. Problem: Find degree of homogeneous function $f(x,y) = \frac{x}{y} + g\left(\frac{y}{x}\right)$. Check homogeneity: $f(tx, ty) = \frac{tx}{ty} + g\left(\frac{ty}{tx}\right) = \frac{x}{y} + g\left(\frac{y}{x}\right) = f(x,y)$, so degree $=0$. Final answers: 1. (b) $-\frac{f_x}{f_y}$ 2. (a) $y x^{y-1}$ 3. (a) 0 4. (c) -2 5. (a) 0 6. Tangent line: $x=0$ 7. Degree: 0