hi everyone today we're going to talk about how to prove the derivatives of inverse hyperbolic
identities to complete this problem we'll use algebra to manipulate the derivative equation
and then simplify as much as possible to prove the derivative let's take a look in this particular
problem we've been asked to show that the derivative of inverse hyperbolic sine is equal
to one divided by the square root of one plus x squared and normally this would be a definition
or an identity that we've given and we'd take it for granted but in this case we need to
prove that this is true one of formulas we're going to use is a very common identity that
we know which is hyperbolic cosine squared root of x minus hyperbolic sine squared of
x is equal to one remember that we need to show that the derivative of inverse hyperbolic
sine here is equal to what we have over here on the right hand side so we're going to hone
in on the derivative of inverse hyperbolic sine the way that we're going to do that is
for now we'll just set y equal to inverse hyperbolic sine and we'll take it from there
so now that we've got in a form of a function we want go ahead and simplify and then take
the derivative so the way that we'll simplify by taking the hyperbolic sine of both sides
so we'll take hyperbolic sine of both sides and we'll get hyperbolic sine of y when we
take hyperbolic sine over here on the right it will cancel with inverse hyperbolic sine
and we'll just be left with x now remember that we're driving toward the derivative of
inverse hyperbolic sine so we want go ahead and take the derivative of what we've got
here with respect to x so this is going to be implicit differentiation and we're going
to take the derivative of both sides with respect to x taking the derivative of the
left hand side remember that the derivative of hyperbolic sine is hyperbolic cosine so
we'll get hyperbolic cosine of y but because we're using implicit differentiation with
respect to x we have to use chain and whenever we take the derivative of y we have to go
ahead and multiple by dy over dx and that's a rule of implicit differentiation so we multiple
by dy over dx on the right hand side taking the derivative of x with respect to x is totally
normal we just get one now we want and go ahead and solve for dy dx by dividing both
sides by hyperbolic cosine y so we'll get dy over dx is equal to one divided hyperbolic
cosine of y what we have now one over hyperbolic cosine of y is the derivative of inverse hyperbolic
sine we just need to get it in the same form as what we have over here on the right hand
side so we've got one in our numerator we need to just modify or change the denominator
from hyperbolic cosine of y to the square root of one plus x squared the way that we'll
do that is with the identity that we've written up here or this formula that's common that
we know we're going to switch this around to solve for hyperbolic cosine of x in this
x with our formula so we'll do that by adding hyperbolic sine squared of x to both sides
so we'll get hyperbolic cosine squared x is equal to one plus hyperbolic sine squared
of x now we want to take the square root of both sides and we'll get hyperbolic cosine
of x is equal to the square root of one plus hyperbolic sine squared of x so we've got
that there we can now plug in what we have on the right hand side for hyperbolic cosine
of y the fact that x and y are different here is irrelevant we have x here because its just
a formula so what we get when we plug that in is one over the square root of one plus
hyperbolic sine squared of y and now as you can see all we need is a way to simplify hyperbolic
sine squared of y the way that we do that is by taking this step right here and squaring
both sides so if we square both sides we'll get hyperbolic sine squared of y is equal
to x squared we can see here that we have hyperbolic sine squared of y and hyperbolic
sine squared y so we can go ahead and make the substitution for x squared and get one
divided the square root of one plus x squared and that's it we've shown that the derivative
of inverse hyperbolic sine is one divided by the square root of one plus x squared and
proven that identity so I hope you found that video helpful if you did like this video down
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