How do you find the instantaneous rate of change of #f(x)=(x^2-2)/(x+4)# at #x=-1#?

To find the instantaneous rate of change of ( f(x) = \frac{{x^2 - 2}}{{x + 4}} ) at ( x = -1 ), you can use the formula for the derivative of a function. The derivative of ( f(x) ) with respect to ( x ) gives the instantaneous rate of change at any given point.

First, find the derivative of ( f(x) ) using the quotient rule, then evaluate it at ( x = -1 ) to find the instantaneous rate of change.

( f'(x) = \frac{{(x+4)(2x) - (x^2 - 2)(1)}}{{(x+4)^2}} )

( f'(x) = \frac{{2x^2 + 8x - (x^2 - 2)}}{{(x+4)^2}} )

( f'(x) = \frac{{x^2 + 8x + 2}}{{(x+4)^2}} )

Now, plug in ( x = -1 ) to find the instantaneous rate of change at that point:

( f'(-1) = \frac{{(-1)^2 + 8(-1) + 2}}{{(-1+4)^2}} )

( f'(-1) = \frac{{1 - 8 + 2}}{{3^2}} )

( f'(-1) = \frac{{-5}}{{9}} )

So, the instantaneous rate of change of ( f(x) ) at ( x = -1 ) is ( -\frac{5}{9} ).