The slope of a curve at a certain point is the same as the slope of the line that is tangent to the curve at that point (touches the curve at ONE point and does not cross over the curve). This is also the derivative (the instantaneous rate of change of a function). Therefore, to find the slope of a curve at point "P", you can take the derivative of the equation and plug in the X and Y values of point P on the curve f(x). For example:

P=(6,4)

f(x)=-x2+9x-14

f'(x)=-2x+9..... (you get this using the formula f'(x)=nx^(n-1))

...SO you then find the slope of the curve at (6,4) by finding the derivative at P:

f'(6)=-2(6)+9=-3

...This means that the slope of the curve f(x) at (6,4) is -3.

Another way to discuss this concept is in terms of limits with the difference quotient (seen below). This is used to find the slope of the secant line, but as the "delta x" gets smaller, the secant line becomes increasingly similar to the tangent line. The slope of the tangent line is also known as the limit.

When a curve has a tangent of zero, there is a maximum or minimum, or the graph is horizontal. There is no tangent of a curve (the function is not differentiable) if the function is not continuous (the limit of f(x) as x approaches a from the right does not equal the limit of f(x) as x approaches a from the left). There is a vertical tangent if and only if  . Vertical tangents usually (but not always) occur at vertical asymptotes. 

An applet demonstrating the slope of a curve at a point: calculusapplets.com/derivpoint.html

SOME OPPORTUNITIES TO ENRICH THE GARDEN THAT IS YOUR MIND...

(1.) The slope of the curve  (y^3)-x(y^2)=4 at the point where y=2 is

       (A) -2

       (B) 1/4

       (C) -1/2

       (D) 1/2

       (E) 2

(2.) The slope of the curve (y^2)-xy-3x=1 at the point (0,-1) is

       (A) -1

       (B) -2

       (C) 1

       (D) 2

       (E) -3

 (3.)