1. **Define Homogeneous Function:** A function $f(x,y)$ is called homogeneous of degree $n$ if for all $t > 0$, it satisfies $$f(tx, ty) = t^n f(x,y).$$ 2. **Modified Euler’s Theorem Statement:** If $f(x,y)$ is a homogeneous function of degree $n$, then $$x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x,y).$$ 3. **Find $f_x(1,3)$ for $f(x,y) = x^2 y + xy^2$:** - Differentiate $f$ partially with respect to $x$: $$\frac{\partial f}{\partial x} = 2xy + y^2.$$ - Substitute $x=1$, $y=3$: $$f_x(1,3) = 2 \times 1 \times 3 + 3^2 = 6 + 9 = 15.$$ 4. **Find $\frac{\partial f}{\partial x}(1,2)$ for $f(x,y) = x^3 + y^3 - 3xy$:** - Differentiate $f$ partially with respect to $x$: $$\frac{\partial f}{\partial x} = 3x^2 - 3y.$$ - Substitute $x=1$, $y=2$: $$\frac{\partial f}{\partial x}(1,2) = 3 \times 1^2 - 3 \times 2 = 3 - 6 = -3.$$ 5. **Find $\frac{dy}{dx}$ if $y \sin x = x \cos y$:** - Differentiate both sides implicitly: $$\frac{d}{dx}(y \sin x) = \frac{d}{dx}(x \cos y).$$ - Using product and chain rules: $$y \cos x + \sin x \frac{dy}{dx} = \cos y - x \sin y \frac{dy}{dx}.$$ - Group $\frac{dy}{dx}$ terms: $$\sin x \frac{dy}{dx} + x \sin y \frac{dy}{dx} = \cos y - y \cos x.$$ - Factor $\frac{dy}{dx}$: $$\frac{dy}{dx}(\sin x + x \sin y) = \cos y - y \cos x.$$ - Solve for $\frac{dy}{dx}$: $$\frac{dy}{dx} = \frac{\cos y - y \cos x}{\sin x + x \sin y}.$$