Differentiate sinh-1 (tan(x))

Question

Differentiate sinh−1 (tan(x))

 

 

ANSWER:  |sec(x)|

practice test, #54

Answer 1

Helpful Identities:

1 + tan²x = sec²x

cosh²x = 1+ sinh²x

Set sinh^-1(tan x) equal to y.

Then sinh y is tan x; differentiate both sides of sinh y = tan x to reach cosh y(dy/dx) = sec²x.

Rewrite cosh y(dy/dx) = sec²x as dy/dx = (1/cosh y) sec²x.

Then dy/dx equals sec²x divided by √(1 + sinh²y).

With sinh y equal to tan x, obtain dy/dx or d[sinh^-1(tan x)]/dx

equals sec²x divided by √(1 + tan²x) or sec²x/√sec²x or sec²x/sec x equal to |sec x|.