Differentiating x² + y² = 25 implicitly gives 2x + 2y(dy/dx) = 0. What is dy/dx?
Ady/dx = x/y
Bdy/dx = -x/y
Cdy/dx = -x
Ddy/dx = 2x
Solve 2x + 2y(dy/dx) = 0 for dy/dx: subtract 2x to get 2y(dy/dx) = -2x, then divide by 2y to get dy/dx = -x/y. The chain rule is the key step that produces 2y(dy/dx) rather than just 2y when differentiating y² — because y is a function of x, not a constant.
Question 2 True / False
When differentiating x³ + y³ = 1 implicitly, the derivative of y³ with respect to x is 3y².
TTrue
FFalse
Answer: False
Because y is a function of x, the chain rule applies: d/dx[y³] = 3y² · dy/dx. The factor dy/dx must be included. Writing just 3y² would be correct only if y were an independent variable, not a function of x. Omitting dy/dx when differentiating terms in y is the defining error of implicit differentiation.
Question 3 Short Answer
Why is implicit differentiation useful for a curve like x² + y² = 25, even though you could solve for y explicitly?
Think about your answer, then reveal below.
Model answer: Solving for y produces two functions (y = ±√(25 − x²)) that must be differentiated separately. Implicit differentiation handles the entire curve in one calculation and works even when y cannot be isolated algebraically.
Implicit differentiation treats y as an unspecified function of x and applies the chain rule to find dy/dx without needing an explicit formula. This is essential for curves like x⁵ + y⁵ = xy where isolating y is algebraically impossible, and more efficient for curves that split into multiple branches.