Related Rates Related rates is a method for finding derivatives of functions with respect to other functions. Related rates are a slight twist on what we did with implicit derivatives earlier, where we found thederivative of y with respect to x. Now, instead of taking the derivative with respect to x, we'll take the derivative with respect to t. Forexample, we'll use equations that represent volume and surface area and use them to solveproblems relating to time. We will take a derivative with respect to time. Let us examine how this works. To begin, let me provide some background on what a related rate is. Up to this point, we've been taking derivatives with respect to x. And now I want to take youthrough a few examples of taking derivatives with respect to other variables. Let us consider the derivative of x cubed. We have said that this is 3x squared. Remember that thevariable may be changed to d over dt t cubed, which is equivalent to 3t squared. It is basically the rate of change of this function with respect to this variable. Whenever they are thesame, there is no real change. We then discussed implicit derivatives, beginning with the derivative of y cubed with respect to x.That was y multiplied by itself three times, or y cubed. We multiplied this expression by dy dx, thederivative of y with respect to x because the chain rule required us to do so. So I could, again, change the letter from x to t and rewrite the equation. Let's say that y is a function of time, t. That would be the derivative of y with respect to t, or d overdt. d over dt is equal to y cubed, where t represents time. This would mean that the derivative of ywith respect to t is 3y squared over dt. It's an application of implicit derivatives. Basically, it's just a way of expressing the derivative of afunction in terms of the function's implicit value. It can be changeable. Let's suppose that r represents the radius of a sphere. Then we're looking for the rate at which thesphere's radius changes with respect to time. That would be 3 r2 dr dt.
When examining related rates, we are usually dealing with an equation that describes a real-worldprocess. Let us assume that the volume of a cube is side cubed. To find the derivative of volume withrespect to time, we must first determine how the volume changes over time. Or it changes withrespect to time. The function v as a function of t is being cubed. We can write the function that way, but it'scumbersome. So we write it as a standard geometric equation, with the derivative of t implicitly included. Then wetake the derivative of v3. This is an application of the chain rule, because v3 is cubed. We write theequation as the normal geometric function you are used to, and we differentiate it. The derivative ofv is implied to be 1. But then the derivative of s cubed is equal to 3s squared times ds dt.