Using implicit differentiation, find dy/dx if x² + y² = 25.

Question

Accepted Answer

-x/y. The correct answer is -x/y. Differentiating both sides with respect to x: 2x + 2y(dy/dx) = 0. Solving for dy/dx: 2y(dy/dx) = -2x, so dy/dx = -2x/(2y) = -x/y. This makes geometric sense: the circle x² + y² = 25 has tangent lines with slopes -x/y at any point (x, y).

The expression x/y has the wrong sign; since the circle curves downward in the first quadrant, the slope should be negative there, requiring the negative sign. The expression -y/x inverts the fraction; this error comes from incorrectly solving the algebraic equation after differentiation. The expression -2x/2y is actually equal to -x/y and is technically correct, but it is not fully simplified. However, if presented as a distinct choice, students should simplify to recognize it equals the correct answer.

Implicit differentiation is essential for the CLEP exam because many curves cannot be written as y = f(x). The technique treats y as a function of x and applies the chain rule whenever y appears. Common applications include circles, ellipses, and other conic sections. Students should practice solving for dy/dx after differentiating and simplifying the result.

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