How do I calculate the rate of change of something for which I don't have an equation?

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This kind of question is an application of differentiation. If t represents time, any derivative with respect to t is a rate of change. For example, if h represents the depth of water in a cubicle container, dh/dt is the change in depth over time, or the rate of change of the depth in other words. If the question does not give you an equation that directly relates the thing you want the rate of change of to time, it will often give you an equation that relates a different quantity to time. For example, it might ask you to calculate the rate of change of volume of water in the container, but not give you an equation for volume in terms of time, and so one differentiation won't suffice. I find it really helpful to write out what I'm trying to calculate and how I could do so with the given terms. Let's say we need to calculate the rate of change of volume of water in the container. We are given that the rate of change of depth (dh/dt) is 10m/s, and that the container is a cuboid with dimensions of 50x50xh m. We want dV/dt. We have dh/dt, so if we write this out we know that dV/dt = (dh/dt)*(some other derivative). If you treat derivatives as fractions (which you normally can), you can see that in order cancel out dh, the other derivative must be dV/dh. We know that the equation for volume in terms of h for a cuboid of dimensions 50x50xh is simply V=2500h. Differentiate this and you get dV/dh=2500. All that's left to do is to multiply the two derivatives together, so dV/dt=10*2500 = 25000. So the rate of change of volume is 25000m^3/s.

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