If you're like me and don't remember hyperbolic identities (especially involving inverse functions) off the top of your head, recall the definitions of the hyperbolic cosine and sine:
Then differentiating yields
so that by the chain rule, if
then
Now, let , so that (•) .
Recall that
and so the derivative of tanh(<em>x</em>) is
where the last equality follows from the hyperbolic Pythagorean identity,
Differentiating both sides of (•) implicitly with respect to <em>x</em> gives
So, the derivative we want is the somewhat messy expression
and while this could be simplified into a rational expression of <em>x</em>, I would argue for leaving the solution in this form considering how <em>y</em> is given in this form from the start.
In case you are interested, we have
and you can instead work on differentiating that; you would end up with
The domain is all the x values so {2,-4, 1, 2, -1}
The range is all the y values so {1, 5, 7, -3, 2}
Yes, it is a function because each item in the domain matches one item in the ranges.
Answer:
7
Step-by-step explanation:
Simplify the following:
13 (-5) - 8 (-9)
13 (-5) = -65:
-65 - 8 (-9)
-8 (-9) = 72:
-65 + 72
-65 + 72 = 7:
Answer: 7
Answer:
A
Step-by-step explanation:
Remark
f and d are equal (and acute) because they are corresponding angles.
82 and f are supplementary, so we can find f
Finding f
f + 82 = 180
f = 180 - 82
f = 82
Finding d
d = f
d = 82