121616949-math.133 - 6.2 Related Rates 119 x to mean dx/dt...

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6.2 Related Rates 119 ˙ x to mean dx/dt goes back to Newton and is still used for this purpose, especially by physicists.) If y is written in terms of x , i.e., y = f ( x ), then this is easy to do using the chain rule: ˙ y = dy dt = dy dx · dx dt = dy dx ˙ x. That is, find the derivative of f ( x ), plug in the value of x at the instant in question, and multiply by the given value of ˙ x = dx/dt to get ˙ y = dy/dt . EXAMPLE 6.13 Suppose an object is moving along a path described by y = x 2 , that is, it is moving on a parabolic path. At a particular time, say t = 5, the x coordinate is 6 and we measure the speed at which the x coordinate of the object is changing and find that dx/dt = 3. At the same time, how fast is the y coordinate changing? Using the chain rule, dy/dt = 2 x · dx/dt . At t = 5 we know that x = 6 and dx/dt = 3, so dy/dt = 2 · 6 · 3 = 36. In many cases, particularly interesting ones, x and y will be related in some other way, for example x = f ( y ), or F ( x, y ) = k , or perhaps F ( x, y ) = G ( x, y ), where F ( x, y ) and G ( x, y ) are expressions involving both variables.
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