Which statement correctly describes dy/dx for a parametric curve in terms of t?

Which statement correctly describes dy/dx for a parametric curve in terms of t?

• A. dy/dx = dx/dt / dy/dt
• B. dy/dx = dy/dt / dx/dt
• C. dy/dx = dy/dt
• D. dy/dx = dx/dt

Answer: (B)

When x and y depend on a common parameter t, the slope of the curve at a point is how y changes with respect to t divided by how x changes with respect to t. In formula form, dy/dx = (dy/dt) / (dx/dt), as long as dx/dt ≠ 0. This follows from the chain rule: dy/dx = (dy/dt) / (dx/dt) because dy/dx = (dy/dt) / (dx/dt) when each variable is viewed as a function of t. The reciprocal would describe the rate dx/dy, not dy/dx, so it’s not the correct expression. For intuition, consider x = t^3 and y = t^2. Then dy/dt = 2t and dx/dt = 3t^2, giving dy/dx = (2t)/(3t^2) = 2/(3t). As t approaches 0, the slope grows without bound, indicating a vertical tangent at that point. Note that if dx/dt = 0, the slope is undefined (a vertical tangent) unless dy/dt is also 0, in which case you’d need a closer look or a reparameterization.

When x and y depend on a common parameter t, the slope of the curve at a point is how y changes with respect to t divided by how x changes with respect to t. In formula form, dy/dx = (dy/dt) / (dx/dt), as long as dx/dt ≠ 0.

This follows from the chain rule: dy/dx = (dy/dt) / (dx/dt) because dy/dx = (dy/dt) / (dx/dt) when each variable is viewed as a function of t. The reciprocal would describe the rate dx/dy, not dy/dx, so it’s not the correct expression.

For intuition, consider x = t^3 and y = t^2. Then dy/dt = 2t and dx/dt = 3t^2, giving dy/dx = (2t)/(3t^2) = 2/(3t). As t approaches 0, the slope grows without bound, indicating a vertical tangent at that point.

Note that if dx/dt = 0, the slope is undefined (a vertical tangent) unless dy/dt is also 0, in which case you’d need a closer look or a reparameterization.