Derivatives of parametric functions (or whatevs)

In summary, the conversation is about finding the derivative of a surface with two parametric equations, x(t,s) and y(t,s), in terms of dx/dt, dx/ds, etc. The person asking the question is confused about how to calculate the derivative with two parameters and gives an example of using height as x and north/south as t and s. They are unsure of the meaning of dy/dx in this scenario.
  • #1
variety
22
0
Quick question. This is kind of embarrassing actually. Suppose I have functions x(t,s) and y(t,s) (say they're parametric equations of a surface of something) and I want to know what dy/dx is. Specifically, I have x and y in terms of the parameters, which are kind of complicated functions, and I want to know what that derivative is in terms of dx/dt, dx/ds, etc.

I know if they were functions of just 1 parameter, then it would just be dy/dx = (dy/dt)/(dx/dt), but with 2 I'm confused. Can you guys help me out?!?
 
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  • #2
hi variety! :smile:
variety said:
Suppose I have functions x(t,s) and y(t,s) … and I want to know what that derivative is in terms of dx/dt, dx/ds, etc.

I know if they were functions of just 1 parameter, then it would just be dy/dx = (dy/dt)/(dx/dt), but with 2 I'm confused.

does it make any sense?

suppose x is height, and t and s are north and south …

then x is a hill, and there are contour lines along which x does not change, but y does change …

what could be the meaning of dy/dx ? :confused:​
 

FAQ: Derivatives of parametric functions (or whatevs)

What are derivatives of parametric functions?

Derivatives of parametric functions are mathematical expressions that describe the rate of change of a parametric equation with respect to its independent variable. In simpler terms, it is the slope or rate of change of a curve at a specific point.

Why are derivatives of parametric functions important?

Derivatives of parametric functions are important because they allow us to study and analyze the behavior of a curve in terms of its rate of change. This is useful in many fields such as physics, engineering, and economics, where understanding how a system changes over time is crucial.

How do you find derivatives of parametric functions?

The standard method for finding derivatives of parametric functions is to use the chain rule, where the derivatives of the x and y components of the parametric equation are multiplied by the derivative of the parameter with respect to the independent variable.

Can derivatives of parametric functions be negative?

Yes, derivatives of parametric functions can be negative. This indicates that the function is decreasing at that point, or that the slope of the curve is negative.

What is the physical interpretation of derivatives of parametric functions?

The physical interpretation of derivatives of parametric functions is that they represent the instantaneous rate of change of a physical quantity. For example, if the parametric equation represents the position of an object over time, then its derivative would represent the object's velocity at a specific time.

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