Derivatives

[Lizzie]

OK, I am supposed to find the derivatives of the following function:

f(x)= (tan(x)-1)/sec(x)

This is what I've done:

I know that you have to take the bottom times the derivative of the top, minus the top times the derivative of the bottom all over 2 times the bottom. So, this is what I get...

(sec(x))(sec,²(x)) - (tan(x)-1)(sec(x)tan(x)) All over (sec(x))²

I don't quite know where to go from there.

 

[Gene]

Rather than jumping into it as-is you might want to consider
(tan(x)-1)/sec(x) =
((sin(x)/cos(x))-1)/sec(x)
((sin(x)-cos(x)))/cos(x)*cos(x) =
sin(x)-cos(x)
Your answer should equal the derivitive of that but unless you insist I'll avoid prooving it.

 

[soroban]

Hello, Lizzie!

I am supposed to find the derivatives of the following function:

f(x) = [tan(x) - 1]/sec(x)

This is what I've done:

I know that you have to take the bottom times the derivative of the top,
minus the top times the derivative of the bottom
all over the square of the bottom.

So, this is what I get:

. . . sec(x)·sec²(x) - [tan(x) - 1]·sec(x)·tan(x)
. . .-------------------------------------------------- . . . . . correct! . Nice work!
. . . . . . . . . . . . . . . sec²(x)

I don't quite know where to go from there.

Just multiply it out and look for some 'identity' to simplify it.
. . (But some problems just don't simplify at all!)

The numerator is: .sec³(x) - sec(x)·tan²(x) + sec(x)·tan(x)

. . Factor: .sec(x) [sec²(x) - tan²(x) + tan(x)]

There is an identity: .sec²θ - tan²θ .= .1

. . So we have: .sec(x) [1 + tan(x)]

. . . . . . . . . . . . . sec(x) [1 + tan(x)] . . . . 1 + tan(x)
The fraction is: . ----------------------- . = . -------------
. . . . . . . . . . . . . . . . . sec²(x) . . . . . . . . . . sec(x)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

I just saw something absolutely silly! . . .

. . . . . . . . . . . . . . . . . . . sin(x)
. . . . . . . . . . . . . . . . . . . -------- .- .1
. . . . . . . tan(x) - 1 . . . . cos(x)
f(x) . = . ------------ . = . ---------------
. . . . . . . . sec(x) . . . . . . . . 1
. . . . . . . . . . . . . . . . . . . .---------
. . . . . . . . . . . . . . . . . . . . cos(x)


Multiply top and bottom by cos(x): . f(x) . = . sin(x) - cos(x)

. . and the derivative is: .f '(x) .= .cos(x) + sin(x)

Ha! . . . We could have done it in 3.7 seconds!

[Edit: Gene beat me to it!]

 

[Lizzie]

Wow, thanks guys. I really appreciate your help. I guess it would have helped if I had remembered the identities, huh? lol