Finding the derevative

Question

Find the derivative of y= coth^-1(√(x^2+1))

Answer 1

***We will use the formula: d/d𝑥(coth⁻¹𝑥)=1/(1−𝑥²).***

y= coth^-1(√(x^2+1))

Find y'.

This is a chain rule problem with three nested functions y = f(g(h)).

Then y' = f'(gh) *g'(h)* h'.

f = coth⁻¹ (x)

g = √x

h = x²+1

f'(gh) = 1/(1−(√(x²+1)²) = -1/x²

g'(h) = 1/(2√(x²+1))

h' = 2x

y' = (-1/x²) (1/(2√(x²+1)))*2x

simplify:

y' = -1/(x(√x²+1))