What is the derivative of sec xsecx?

2 Answers
Jun 5, 2016

It is sin(x)/cos(x)^2sin(x)cos(x)2.

Explanation:

sec(x)=1/cos(x)sec(x)=1cos(x)

So we want to calculate

d/dx1/cos(x)=d/dx(cos(x)^-1)ddx1cos(x)=ddx(cos(x)1)

for the chain rule this is equal to

d/dx(cos(x)^-1)=-cos(x)^-2*d/dxcos(x)ddx(cos(x)1)=cos(x)2ddxcos(x)

=-1/cos(x)^2*(-sin(x))=1cos(x)2(sin(x))

=sin(x)/cos(x)^2=sin(x)cos(x)2

or, if you prefer, it is

=tan(x)sec(x)=tan(x)sec(x).

Jun 5, 2016

d/dxsecx=secxtanxddxsecx=secxtanx

Explanation:

To find the derivative of secant, we could either use the limit definition of the derivative (which would take a very long time) or the definition of secant itself:
secx=1/cosxsecx=1cosx

We know d/dxcosx=-sinxddxcosx=sinx - keep that in mind because we're going to need it.

Our problem is:
d/dxsecxddxsecx

Since secx=1/cosxsecx=1cosx, we can write this as:
d/dx1/cosxddx1cosx

We can find this derivative using the quotient rule:
d/dxu/v=(u'v-uv')/v^2

In our case, u=1->u'=0 and v=cosx->v'=-sinx:
d/dx1/cosx=((0)(cosx)-(1)(-sinx))/(cosx)^2
=sinx/cos^2x

This is equivalent to:
1/cosx*sinx/cosx

Because 1/cosx=secx and sinx/cosx=tanx, this is:
secxtanx

Therefore,
d/dxsecx=d/dx1/cosx=secxtanx

Pro Tip
You can use this method to find the derivative of cscx and cotx by recognizing cscx=1/sinx and cotx=1/tanx and using the quotient rule, as we did above. This way, you only have to memorize the derivatives of sine, cosine, and tangent - you can derive the other three.