How do you find the rate of change?

The rate of change tells us how one thing changes compared to another. For example, if you're looking at how far a car travels over time, the rate of change would be its speed—how many meters the car moves every second.

First we need to know how much each of these things changes. You do this by subtracting the starting value from the ending value. For example, if a car moves from 10 meters to 30 meters in 4 seconds, the change in distance is 20 meters, and the change in time is 4 seconds.

Then you divide the change in distance by the change in time to get the rate of change. In this case, the car is moving at 5 meters per second. This means every second, the car travels 5 meters.

Hope this helps :)

in general, finding the gradient of any function will give you the rate of change of it at a certain point. if you don't know calculus, then draw a tangent, pick two coordinates and then find the change in y divided by the change in x. if you do know calculus, differentiating the function once and then substituting in the x coordinate of the point of interest will give you the gradient at that point

Hi Vickie!

The rate of change of a system can be found in several different ways depending on the particulars of the question. For this answer, I will assume you're comfortable with single differentiation, and we will talk about some of the cases where there are several derivatives at play! If you would like help with single derivatives for rates of change, then please follow this reply up with another question :)

The crucial thing to ask ourselves is on what conditions (also known as "independent variables") does whatever we are measuring (our "dependent variable") depend on? Here is a quick scheme and example for the rate of change of a dependent variable which depends on two factors:

Let V be the volume of a cuboid with a square base with sides of length x meters, and a height of y meters. Then from geometry we know the volume is given by

V(x,y) = y*(x^2),

where we write V(x,y) because V depends on both x and y, and x^2 is typed notation for x squared. Now if we change x by a small amount, how does this affect V? This is of course a lengthier way of asking: what is the rate of change of V with respect to x? We can calculate this with a derivative, which measures precisely this! If we remember our differentiation, we get:

The rate of change of V with respect to x is given by:

dV/dx = 2xy.

The rate of change of V with respect to y is given by:

dV/dy = x^2.

Now, what if both x and y are changing at the same time? Then we need to consider a global rate of change (which we will label dV/dt), and this is where things get a bit tricky. In summary, we will add together both rates of change! However, we need to be careful to account for how fast x and y are both changing.

Let's pretend that x changes at a rate of 1meter per second, and y changes at a rate of 2meters per second. Then if we add the rates of change, we get:

dV/dt = x^2 + 2xy,

but this is not the whole story! As y is changing at double the speed than x, intuitively we should make sure that the term dV/dy has double the effect on the global rate of change. The correct answer is therefore:

dV/dt = x^2 + 4xy.

This example illustrates an intuitive approach to how rates of change work with several moving parts. The topic called implicit differentiation then takes these ideas and makes them much more automatic! In summary,

The global rate of change is found by multiplying the individual rate of change of each variable by how fast that variable is changing, and adding all these terms together for all independent variables.

Hope this helps!

Alberto

Hi Vickie,

The rate of change is the amount of change that occurs per unit time. It could be metres per second, energy per hour, or pence per day.

Calculate the change that has happened, and divide it by the amount of time that has passed. Make sure that the units you are using, are the units you want in your answer - for example - if the change was 5m and it took 1minute, but you need to express the answer as cm per second, then you will need to change the 5metres into centimetres and the 1minute into seconds. Answer: 500/60 = 8.33cm/s

Good luck!