Do you mean
cos(x+∆h)
If so, then
cos(x+∆h) = cos(x)cos(∆h) - sin(x)sin(∆h)
This symbol ∆ is normally used to indicate a "small amount of" or a difference.
It is used when figuring out where derivative come from. Since the derivative is the instantaneous slope, you try to find how fast a function grows, when x is increased by a small amount (for example, by ∆h).
The increase is the difference
cos(x+∆h) - cos(x)
And the slope is
[cos(x+∆h) - cos(x)] / ∆h
The trick is to find the limit as ∆h gets smaller and smaller.
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As for the limit itself, it is found in this way:
[cos(x+∆h) - cos(x)] / ∆h
can be rewritten:
[cos(x)cos(∆h) - sin(x)sin(∆h) - cos(x)] / ∆h
rearrange the order:
[cos(x)cos(∆h) - cos(x) - sin(x)sin(∆h)] / ∆h
Split the numerator:to separate the cos from the sin:
[cos(x)cos(∆h) - cos(x)] / ∆h - [sin(x)sin(∆h)] / ∆h
The first term:
we know cos(0) = 1
as ∆h approaches 0, cos(∆h)=1, leaving us with
cos(x)*1 - cos(x) = cos(x) - cos(x) = 0
The first term is gone.
The second term:
as ∆h gets smaller and smaller, we find that sin(∆h) = ∆h
(for angles in radians)
This is a trick used in astronomy, where angles can be very small.
This means that the limit, as ∆h approaches 0, is
sin(∆h) / ∆h = 1
Leaving us with
-[sin(x)sin(∆h)]/∆h = -sin(x)*1 = -sin(x)