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

Question

3173 views around the world

You can reuse this answer
Creative Commons License

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:

$dy/dx = 5/coshy$

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

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