Answer: dy/dx = y(y - x ln y)/(x(x - y ln x)).
- A dy/dx = y(y - x ln y)/(x(x - y ln x))
- B dy/dx = y/x, obtained by treating the exponents as if they cancelled directly
- C dy/dx = x/y, obtained by inverting the y/x ratio without full log differentiation
- D dy/dx = ln(y/x), mistakenly equating the derivative with the log of the ratio
Correct answer: A. dy/dx = y(y - x ln y)/(x(x - y ln x))
Explanation: Taking ln: y ln x = x ln y. Differentiating implicitly: (dy/dx)ln x + y/x = ln y + x(1/y)(dy/dx). Solving for dy/dx gives dy/dx = y(y - x ln y) / (x(x - y ln x)).
f(x)=|x| is perfectly continuous at x=0 (no break, no jump), but the curve has a sharp corner there: approaching from the left gives slope -1 and from the right gives slope +1 - since these one-sided derivatives disagree, the function is not differentiable at that point.
Concept context
When a function has no breaks (continuity) and when it has a well-defined slope (differentiability), plus rules for differentiating implicit, inverse trig, exponential, log, and parametric functions.