Question

If \( x^{2}y - xy^{2} + x^{3} - y^{3} = 0 \), then \( \frac{dy}{dx} \) at the point (1, 1) is

• A. \( 1 \)
• B. \( 0 \)
• C. \( -1 \)
• D. Does not exist

Hint:

The partial derivative formula \( \frac{dy}{dx} = -\frac{f_x}{f_y} \) is significantly faster and less prone to algebraic layout errors than differentiating term-by-term and rearranging elements manually.

Answer 1

The Correct Option is A

Step 1: Use the implicit derivative shortcut.
For a curve $f(x,y)=0$, \[ \frac{dy}{dx}=-\frac{\partial f/\partial x}{\partial f/\partial y} \] Here $f(x,y)=x^2y-xy^2+x^3-y^3$.
Step 2: Differentiate treating $y$ as constant.
\[ \frac{\partial f}{\partial x}=2xy-y^2+3x^2 \]
Step 3: Differentiate treating $x$ as constant.
\[ \frac{\partial f}{\partial y}=x^2-2xy-3y^2 \]
Step 4: Plug in $(1,1)$ for the $x$-part.
\[ \frac{\partial f}{\partial x}\Big|_{(1,1)}=2(1)(1)-1+3=4 \]
Step 5: Plug in $(1,1)$ for the $y$-part.
\[ \frac{\partial f}{\partial y}\Big|_{(1,1)}=1-2-3=-4 \]
Step 6: Combine.
\[ \frac{dy}{dx}=-\frac{4}{-4}=1 \] \[ \boxed{1} \]