# Derivative of Trigonometric Functions

## Homework Statement

d/dx(sec(x)/1+tan(x)
Evaluate at x=∏/6

## The Attempt at a Solution

((1/cos(x))(1+tan(x))-(sec(x))(sin(x)/cos(x)))/(1+tan(x))^2

((1/cos(x))-(sec(x))(sin(x)/cos(x)))/(1+tan(x))

I used the quotient rule and reduced what I could. Have I done this right and can I go any further before evaluating? Thank you.

vela
Staff Emeritus
Homework Helper
No, that's not correct. Applying the quotient rule involves taking some derivatives. It doesn't appear you've done that.

Of course I have. 1/cos(x) is the derivative of sec(x) and (sin(x)/cos(x)) is the derivative of (1+tan(x)) are they not?

vela
Staff Emeritus
Homework Helper
No, they're not.

Right, they are not. Hahaha. Don't mind my red cheeks.

=((1+tanx)(secxtanx)-(secx)(sec^2x))/(1+tanx)2
=(secxtanx+secxtan^2x-sec^3x)/(1+tanx)2

Is this looking any closer to a correct answer?

arildno
Homework Helper
Gold Member
Dearly Missed
Yup!

Is this as far as it goes? Can it be simplified any more?

Mark44
Mentor
Of course I have. 1/cos(x) is the derivative of sec(x) and (sin(x)/cos(x)) is the derivative of (1+tan(x)) are they not?

I realize that you already recognized that these are wrong, but I thought it worth mentioning why they were wrong.

sec(x) = 1/cos(x) - This is an identity and is how the secant function is defined. It has nothing to do with derivatives.

tan(x) = sin(x)/cos(x) - This is also an identity and is how the tangent function is defined. This is unrelated to derivatives.

Yes, I was simply confusing myself. As far as my current work, can it be simplified any more?

Mark44
Mentor
Yes, I was simply confusing myself. As far as my current work, can it be simplified any more?
You might be able to, but what's the point? Just go ahead and evaluate it now at x = ##\pi/6##.

((2/√3)(√3/3)+((2/√3)(1/3))-(8/3^(3/2)))/(4/3)+((2√3)/3)
=(2/9+2/(3√3)-8/(3^3/2))/((2(2√3+1))/3√3)

Can anyone tell me if I am on the right track please?

Mark44
Mentor
Can anyone tell me if I am on the right track please?

=((1+tanx)(secxtanx)-(secx)(sec^2x))/(1+tanx)2
=(secxtanx+secxtan^2x-sec^3x)/(1+tanx)2
The above is fine, but what you have below is difficult to read, especially what you have on the right side of the =. Please simplify it.

((2/√3)(√3/3)+((2/√3)(1/3))-(8/3^(3/2)))/(4/3)+((2√3)/3)
=(2/9+2/(3√3)-8/(3^3/2))/((2(2√3+1))/3√3)

Let's try this

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Oh I guess I could change the term in the denominator to 4/3.

Bump.

Mark44
Mentor
In the 3rd line of your attachment, the sign of the last term mysteriously changed.