Find the derivative of h(x)=6arcsec (x/3).

Question

Find the derivative of h(x)=6arcsec (x/3). Please show all steps.

Answer 1

The function arcsec is normally defined to have the range [0,π], where the derivative is always positive.

Therefore

d/du(arcsec(u)) = 1/(|u|√(u²-1))

To calculate your derivative of h(x), use the substitution u = x/3.

That makes

du/dx = 1/3

dh/dx =

d/dx(6×arcsec(x/3)) =

6×d/dx(arcsec(x/3)) =

6×d/du(arcsec(u))×du/dx =

6×(1/(|u|√(u²-1)))×(1/3) =

2/(|u|√(u²-1)) =

2/(|x/3|√(x²/9 - 1)) =

18/(|x|√(x² - 9))