Video Transcript
Find the derivative of the function
𝑦 equals inverse cosh of the square root of 𝑥.
There are a few ways of going about
this problem. But probably the simplest way is to
rearrange the expression to eliminate the inverse cosh and use implicit
differentiation.
Beginning with 𝑦 equals the
inverse cosh of the square root of 𝑥, we can take the cosh of both sides of the
equation to give cosh 𝑦 equals the square root of 𝑥. Recall that when given a function
𝑓 of 𝑦, we can implicitly differentiate this expression with respect to 𝑥. So, d by d𝑥 of 𝑓 of 𝑦 is equal
to the derivative of the function with respect to 𝑦, 𝑓 prime of 𝑦, times d𝑦 by
d𝑥.
Recall that when given a function
𝑓 of 𝑦, we can implicitly differentiate the function with respect to 𝑥. So, d by d𝑥 of 𝑓 of 𝑦 is equal
to the derivative of the function with respect to 𝑦, 𝑓 prime of 𝑦, times d𝑦 by
d𝑥. So, in this case, on the left-hand
side, we have the derivative of cosh 𝑦 with respect to 𝑦, which is sinh 𝑦, times
d𝑦 by d𝑥. On the left-hand side, we just have
the derivative of root 𝑥 with respect to 𝑥, which gives one over two root 𝑥.
Now we can divide both sides of the
equation by sinh 𝑦, giving d𝑦 by d𝑥 equals one over two root 𝑥 sinh 𝑦. Now, since we have 𝑥 in terms of
cosh 𝑦 rather than sinh 𝑦, we can use a hyperbolic identity sinh 𝑦 is identically
equal to the square root of cosh squared 𝑦 minus one to get this right-hand side in
terms of 𝑥. So this right-hand side becomes one
over two root 𝑥 root cosh squared 𝑦 minus one.
Since cosh 𝑦 is equal to root 𝑥,
cosh squared 𝑦 is equal to 𝑥. So this gives d𝑦 by d𝑥 equals one
over two root 𝑥 root 𝑥 minus one.
