Determine the differential for following.

                                     y = t ³ - 4t ² + 7t

Solution

Before working any of these we have to first discuss just what we're being asked to determine here. We described two differentials earlier & here we're being asked to determine a differential.

Therefore, which differential are we being asked to determine? In this type of problem we're being asked to determine the differential of the function. In other terms, dy for the first problem, dw for the second problem & df for the third problem.

Following are the solutions.  Not much to act here other than takes a derivative & don't forget to add on the second differential to the derivative.

                                      dy = (3t ² - 8t + 7 ) dt

There is a good application to differentials. If we think of ?x as the change in x then  Δy = f ( x + Δx ) - f ( x ) is the change in y equivalent to the change in x.  Now, if Δx is small we can suppose that Δy ≈ dy . Let's see an example of this idea.