Question

if d/dx sec(x) = sec(x)tan(x). What is d/dx (sec^-1(x))?

Answer 1

you want to find d/dx (sec^-1 x) using that formula only?

Answer 2

let
y = sec^-1 x

x= sec y
differentiate both sides w.r.t y,
what u get ?

Answer 3

how do i differentiate both sides?

Answer 4

"d/dx sec(x) = sec(x)tan(x)."

that came by differentiation sec x

so,
d/dy of sec y = ... ?

Answer 5

is it sec(y)tan(y)?

Answer 6

yes!

Answer 7

so,
dx/dy = sec y tan y

now use the fact that
dy/dx = 1/ (dx/dy)
so
dy/dx = ...?

Answer 8

is it 1/sec y tan y

Answer 9

yes
now we just need to find sec y and tany in terms of x

sec y is already = x
can you try to find tan y in terms of x ?

Answer 10

hint :
sec^2 y + tan^2 y =1

Answer 11

is it sec^2(x)-1

Answer 12

that would be tan^2x

we need tan y
tan y = \(\sqrt{\sec^2y-1}\)
with sec y =x
what will be tan y ?

Answer 13

\[\sqrt{x ^{2}-1}\]

Answer 14

yes, so
d/dx (sec^-1 x) = 1/ [x (\(\sqrt {x^2-1}\))]
thats it!

Answer 15

yay thank you for helping me with this problem!!!

Answer 16

welcome ^_^