# Chain Rule Disaster

• Oct 16th 2009, 09:37 PM
chrisf
Chain Rule Disaster
I have a friend who I'm trying to help with calculus. I've been through it already, and I realize we're gonna have to apply the chain rule lots of times... but somewhere we're both screwing up. Could somebody possibly work this problem out (and show each portion)? I'd really appreciated it.

Find the derivative.

sin(sec(tan(cot(csc(cos(x))))))
• Oct 16th 2009, 09:45 PM
Bruno J.
Wolfram Alpha does that pretty well and shows all the steps. Link! (Click "show steps")
• Oct 16th 2009, 09:49 PM
alexmahone
Quote:

Originally Posted by chrisf
I have a friend who I'm trying to help with calculus. I've been through it already, and I realize we're gonna have to apply the chain rule lots of times... but somewhere we're both screwing up. Could somebody possibly work this problem out (and show each portion)? I'd really appreciated it.

Find the derivative.

sin(sec(tan(cot(csc(cos(x))))))

let y=sin(sec(tan(cot(csc(cos(x))))))

dy/dx=cos(sec(tan(cot(csc(cos(x))))))*sec(tan(cot(csc (cos(x)))))*tan(tan(cot(csc(cos(x)))))*sec^2(cot(c sc(cos(x))))*-cosec^2(csc(cos(x)))*-csc(cos(x))cot(cos(x))*-sin(x)
• Oct 16th 2009, 09:50 PM
chrisf
Whoa, that's amazing. Thanks!
• Oct 17th 2009, 02:37 AM
HallsofIvy
Quote:

Originally Posted by chrisf
I have a friend who I'm trying to help with calculus. I've been through it already, and I realize we're gonna have to apply the chain rule lots of times... but somewhere we're both screwing up. Could somebody possibly work this problem out (and show each portion)? I'd really appreciated it.

Find the derivative.

sin(sec(tan(cot(csc(cos(x))))))

The derivative of sin is cos so it is cos(sec(tan(cot(csc(cos(x)))))) times the derivative of sec(tan(cot(csc(cos(x))))).

The derivative of sec is sec*tan so the derivative of that is sec(tan(cot(csc(cos(x)))))tan(tan(cot(csc(cos(x))) )) times the derivative of tan(cot(csc(cos(x)))).

The derivative of tan is \$\displaystyle sec^2\$ so the derivative of that is \$\displaystyle sec^2(cot(csc(cos(x))))\$ times the derivative of cot(csc(cos(x))).

The derivative of cot is \$\displaystyle -csc^2\$ so the derivative of that is \$\displaystyle -csc^2(csc(cos(x)))\$ times the derivative of csc(cos(x)).

The derivative of csc is -csc*cot so the derivative of that is -csc(cos(x))cot(cos(x)) times the derivative of cos(x) which is -sin(x).

Multiply all of those together to get alexmahone's answer.