Understanding Derivatives: Real-World Application Examples

[Iron_Brute]

I'm not sure if this should go in this section or the homework section but I'm having a problem fully understanding what a derivative means in a real world application. My class started moving into basic application problems but I'm not sure what the derivative means. An example of what I mean is a ladder problem in my cal book.
A ladder at 90 degress is sliding horizontally at 2 ft/sec, and in the solution section it says that dx /dt = 2 ft/sec is a given but I don't know why that is a given. In the other example problems any numbers that are "going at a rate of" the solution in the text is saying that that number is the derivative with respect to time, but I don't understand why and can't conceptually understand what that means

The way I understood derivative is that it is the slope of tangent line. So if anyone could explain this to me I'd really appreciate it.

 

[lurflurf]

The derivative is the limit of a ratio
dx/dt=2 ft*sec^-1
usually means x is a distance and t is a time.

 

[gammamcc]

Derivative is the rate of change over an instant of time (hence it's a limit). On the ladder problem, it would be like little speedometers on the corners of the ladder. If you graphed height of latter vs. time, say, the speedometer at any time during the fall comes out the same as the slope of the tangent line on graph at any fixed point in time.

 

[chislam]

Recall the slope of a graph is the change in y / change in x. Thus, if the graph axes are feet and seconds then change in feet / change in seconds is the slope or as more colloquially known in calculus, derivative.