What is the derivative of f(x)=sec−1(x) ?
1 Answer
Process:
First, we will make the equation a little easier to deal with. Take the secant of both sides:
y=sec−1x
secy=x
Next, rewrite in terms of
1cosy=x
And solve for
1=xcosy
1x=cosy
y=arccos(1x)
Now this looks much easier to differentiate. We know that
so we can use this identity as well as the chain rule:
dydx=−1√1−(1x)2⋅ddx[1x]
A bit of simplification:
dydx=−1√1−1x2⋅(−1x2)
A little more simplification:
dydx=1x2√1−1x2
To make the equation a little prettier I will move the
dydx=1√x4(1−1x2)
Some final reduction:
dydx=1√x4−x2
And there's our derivative.
When differentiating inverse trig functions, the key is getting them in a form that's easy to deal with. More than anything, they're an exercise in your knowledge of trig identities and algebraic manipulation.
