Implicit Differentiation

When a car drives on a straight highway with constant acceleration, its velocity is an explicit function of time and gives a linear relationship between time and velocity.

A satellite in circular orbit follows a path described by an implicit function, where x and y are linked together in one equation without isolating a dependent variable.

For the satellite at a given position, the slope shows the instantaneous direction of the motion, and the tangent line shows the velocity vector of the satellite.

To find the slope and tangent, differentiation is applied to the implicit function. To understand the concept of implicit differentiation, consider the equation of a circle.

First, differentiate both sides of the equation with respect to the independent variable. The resulting expression yields the slope of the tangent line.

This slope is then evaluated by substituting the x and y coordinates of the point of tangency.

Finally, the tangent line equation is constructed using the slope and these coordinates, expressed in terms of the original variables.

Similarly, for a moving satellite, at any point, the slope and tangent can be found using the concept of implicit differentiation.