If the rate of change in #x# is #"3 s"^(-1)#, and #(dy)/(dx) = 5#, what is the rate of change in #y#? Is #y# changing faster than #x# or vice versa?

Answer 1
The rate of change, or derivative, of #y# with respect to #x# can be written as
#(dy)/(dx) = 5#,

or

#dy = 5dx#.

We can say that on a non-infinitesimally-small scale,

#Deltay# #~~# #5Deltax#.
Therefore, if #x# changes as #"3 s"^(-1)#, then we let #Deltax = 3#, and
#Deltay = 5xx3 = "15 s"^(-1) > Deltax#,
and we have that #color(blue)(Deltay > Deltax)#.
This means #y# is changing faster than #x#, if we assume that the change in #x# does not itself change, and the change in #y# does not itself change either.
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Answer 2

The rate of change in y is (15 , \text{s}^{-1}). Since (\frac{dy}{dx} > 0), y is changing faster than x.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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