Slope of Tangent Line

Secant line is the straight line joining the function y=ƒ(x) at x and x+Δx. Slope is the rate of change of a straight line. Therefore the rate of change of the secant line is equal to the slope of the secant line. Since the tangent line at x can be approximated by the secant line when applying limit Δx to zero, the derivative can be considered as the slope of tangent line at x.

Direction

The motion of an object usually has both magnitude and direction. In general, the velocity of motion is always tangent to the path of the object motion. Therefore the direction of motion at position x in a circular or curvilinear motion is alway parallel to the tangent line at position x.

Circular Motion Problem

For a circular motion with uniform angular speed ω, the relationship between the displacement in x and the displacement in y can be expressed as a function:

and

The derivative of function with respect to x is

Therefore the derivative of function y with respect to x is a function of x and y. The minus sign means a negative slope. And express the derivative of function with respect to x in term of x, imply:

The plus and minus sign of displacement y represent the upper half circle and the lower half circle of the circular motion. Let the radius r of the circular motion be 5. The derivative, slope of the direction of the object motion is:

Graphically:

Projectile Problem

For parabolic motion of  a projectile with uniform horizontal speed v, the relationship between the displacement in x and the displacement in y can be expressed as a function:

and

The derivative of function with respect to x is

The derivative of function y with respect to x is a function of x only. The minus sign means a negative slope. The derivative, slope of the direction of the object motion is:

Graphically: