3-2-13
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Looking for the derivative.
So the limit as H approaches zero of the quantity three of
the quantity X + H + 5 / X + H minus the quantity three X + 5 /
X.
So the limit as H goes to zero.
We're just going to do some algebra here.
Distribute the three.
So 3X plus three H + 5 / X + H, distributing the negative -3 X
-5 / X.
So the 3X and the -3 X cancel.
So we get the limit as H goes to zero of three H + 5 / X + H -, 5
X all over H Then we're going to separate the three H / H portion
out because three H / H is just three.
The limit as H goes to 03 is 3, so we really need to look at
what happens to that second part.
So the limit is H goes to 0 of 5 / X + H -, 5 / X all over H So
we're going to get a common denominator, and that's going to
be X + H * X.
So we get five X -, 5 times the quantity X + H / X + H * X all
over H So then distributing the -5, the 5X and the -5 X are
going to cancel and we get the limit as H approaches zero of
the -5 H right here of X + H divided by XH.
The H is all now cancel.
So we have that three from before.
Plus the limit is H goes to 0 of -5 / X + H * X.
When H goes to zero, we end up with 3 - 5 / X ^2.
The second part says what happens or what's the slope at
the tangent line X equal to.
So at X equal to, we literally are just going to stick 2IN for
the X back on the derivative 3 -, 5 / 2 ^2 3 -, 5 fourths, 12
fourths -5 fourths, which would give us 7 fourths.