What is the derivative of #y = sinh^(-1)5x#?

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Answer 1 · 2017-05-10T02:13:01

# dy/dx = 5/sqrt(1 + 25x^2)#

Let:

# y = sinh^(-1)5x => sinhy=5x#

Differentiating Implicitly we have:

# coshy dy/dx = 5 #
# :. dy/dx = 5/coshy #

Now using the Hyperbolic Identity:

# cosh^2x-sinh^2x -= 1#

We can write:

# cosh^2x - (5x)^2 = 1#
# :. cosh^2x = 1 + 25x^2#
# :. \ coshx = sqrt(1 + 25x^2) #

So then:

# dy/dx = 5/coshy #

# \ \ \ \ \ \ = 5/sqrt(1 + 25x^2)#