Implicit Differentiation

In classical mechanics, motion is often described through relationships between spatial coordinates and time. A car moving along a straight highway with constant acceleration serves as a simple case where velocity is an explicit function of time. This scenario results in a linear equation, enabling straightforward analysis using basic differentiation techniques.

In contrast, a satellite in circular orbit follows a path defined by an implicit function. The position of the satellite is constrained by the equation of a circle, which links the coordinates x and y without isolating a dependent variable.

Implicit Differentiation for Circular Motion

For a satellite in circular orbit, the position satisfies the equation:

To determine the instantaneous direction of motion—represented by the slope of the tangent line—implicit differentiation is applied. Differentiating both sides with respect to x:

This derivative represents the slope of the tangent line at any point (x,y) on the satellite's path.

Tangent Line Equation

Using the point-slope form, the tangent line at the point (x₁,y₁) is:

This equation provides the direction of the satellite’s velocity vector at a given position. Implicit differentiation thus plays a central role in analyzing the motion of objects constrained by geometric paths, such as satellites in orbit.