Thread: can someone explain to me..

why in this derivative the tan(x) doesn't distribute to the sec(x)

f(x)= sec(x)tan(x)

= sec^2(x) (sec(x)) + (sec(x)tan(x))(tan(x))

= sec^3(x) + tan^2(x) sec(x)

maybe I'm just missing an easy rule, I'm not sure though

Originally Posted by tbenne3

why in this derivative the tan(x) doesn't distribute to the sec(x)

f(x)= sec(x)tan(x)

= sec^2(x) (sec(x)) + (sec(x)tan(x))(tan(x))

= sec^3(x) + tan^2(x) sec(x)

maybe I'm just missing an easy rule, I'm not sure though

Do you mean, why is it not



Yes, that's an equivalent expression

Originally Posted by Archie Meade

Do you mean, why is it not



Yes, that's an equivalent expression

I was wondering why it wouldn't be

(sec(x)tan(x))(tan(x))

= tan(x)sec(x)tan^2(x)

You're thinking too hard. It's simply the distributive property (not) at play.
(a * b) * c = a * (b * c) = a * b * c

In this case sec x * tan x * tan x just ends up being tan^2(x) * sec(x).

Hope that helped!

Originally Posted by experiment00005

You're thinking too hard. It's simply the distributive property (not) at play.
(a * b) * c = a * (b * c) = a * b * c

In this case sec x * tan x * tan x just ends up being tan^2(x) * sec(x).

Hope that helped!

Ok but why is the sec(x) not multiplied by the tan(x).. It looks as if the tan(x) is only distributing to the tan(x) within the parenthesis instead of both sec(x) and tan(x)

Originally Posted by tbenne3

Ok but why is the sec(x) not multiplied by the tan(x).. It looks as if the tan(x) is only distributing to the tan(x) within the parenthesis instead of both sec(x) and tan(x)

You could plug in some values to prove that it doesn't work. Try pi/6.

( sec(pi/6) * tan(pi/6) ) * tan (pi/6) = (2/3) *(sqrt(3)/3) = 2 sqrt(3) / 9

which does not equal (sec(pi/6) *(tan(pi/6) ) * (tan^2(pi/6)) = (2/3)*(1/3)
= 2/9

Originally Posted by tbenne3

why in this derivative the tan(x) doesn't distribute to the sec(x)

f(x)= sec(x)tan(x)

= sec^2(x) (sec(x)) + (sec(x)tan(x))(tan(x))

= sec^3(x) + tan^2(x) sec(x)

maybe I'm just missing an easy rule, I'm not sure though

It looks like you have





This is not the derivative! Gotta watch the distributions, Product Rule :









Does this help?

Originally Posted by Kasper

It looks like you have





This is not the derivative! Gotta watch the distributions, Product Rule :









Does this help?

Yes but why in the third step is the tan(x) only distributed to the tan(x) giving you tan^2(x). Why is it not distributed to the sec(x) also?

Originally Posted by tbenne3

Yes but why in the third step is the tan(x) only distributed to the tan(x) giving you tan^2(x). Why is it not distributed to the sec(x) also?

But it is!

You can also think of it like this:

Say we have , would you agree that



This is the same for , you can just think of the product as being of 3 parts, where two pieces are the same.

We can write it like this too if it helps;



EDIT : Just hit me you might be talking about the tan not distributing to the term on the right. And that is just due to the nature of the product rule.

Just focus on , where for this problem; and .

Just plug away and take derivatives where appropriate.

For



Using the product rule of differentiation,





as



and



This is



or



which has sec(x) as a factor, so it may also be written



When you asked, why is



not equal to



it's because you've brought in an extra tan(x) that does not belong