Differentiating Trigo: 18tan^2(x)sec^2(x)

[DorumonSg]

18tan^2(x)sec^2(x)

Is this correct?

Product Rule I think...

18tan^2(x)sec^2(x) = (18tan^2(x))(2sec^3(x)) + (sec^2(x))(36tan^3(x)) = (36tan^3(1+tan^2(x))) + (36tan^3(1+tan^2(x))) = 2(36tan^3(1+tan^2(x)))

1 more thing is tan x = sec^2(x)? Any 1 has a list of Trigo formulas? Becoz' I am stump here as my course assumes we learned all this stuff from High School, but I didn't attend High School...

 

[LumenPlacidum]

tan(x) is not sec^2(x).

Can you please state your understanding of what the product rule says, what the chain rule says, what the derivative of tan(x) is, and what the derivative of sec(x) is? You're making an error in your calculations, and I want to know where the misunderstanding is or if you just made an arithmetic mistake.

 

[DorumonSg]

Erm... the product rule is when you have 2 parts of an equation like (x)(y), so to different them... u take (x)(y)' + (x)'(y) when ' denotes differentiate it?

Derivative of tan x... I have no idea how to calculate it, but formula say sec^2(x)?

 

[LumenPlacidum]

Yes, the derivative of tan x is sec^2 x, which can be done using tan x = sin x / cos x and doing a product/quotient rule.

Your understanding of the product rule looks good.

In your first step, then you evaluate the derivative of sec^2(x) to be 2sec^3(x)? How did you arrive at that?

 

[DorumonSg]

Meh... I got it mixed up with integration... its supposed to be 2sec(x) right?

 

[LumenPlacidum]

...and then you proceed with the chain rule.

the derivative of f(g(x)) is f'(g(x))*g'(x)

 

[DorumonSg]

Okie. Just one more question, I am sure I am right this time

I've been told to use chain rule to solve this 2+tan^3

So I take 2+tan^3 as my U, where after differentiate I get a 1 because anything to the power of 0 is 1.

The taking 1 x (2+tan^3(x))' = 3tan^2(x)... but it seems my answer is wrong? I am suppose to get 3tan^2(x)(tanx)' or (3tan^2)(sec^2(x))?

And how do u integral U^-1? U can't power rule...

 

[Mark44]

Okie. Just one more question, I am sure I am right this time

I've been told to use chain rule to solve this 2+tan^3

So I take 2+tan^3 as my U, where after differentiate I get a 1 because anything to the power of 0 is 1.

The taking 1 x (2+tan^3(x))' = 3tan^2(x)... but it seems my answer is wrong? I am suppose to get 3tan^2(x)(tanx)' or (3tan^2)(sec^2(x))?

And how do u integral U^-1? U can't power rule...

Your use of terminology is pretty sloppy and unclear; for instance "solve 2 + tan^3" is totally meaningless. I'm assuming that you want to differentiate 2 + tan³(x).

If so, then d/dx(2 + tan³(x)) = d/dx(2) + d/dx(tan³(x)), using the sum rule.

Now use the chain rule to get d/dx(tan³(x)).

Why do you want to integrate? The power rule for integrals says that
\int u^n du~=~ \frac{u^{n + 1}}{n + 1}~for~n~\neq~-1

There's a different formula to use if n = -1.

 

[DorumonSg]

Your use of terminology is pretty sloppy and unclear; for instance "solve 2 + tan^3" is totally meaningless. I'm assuming that you want to differentiate 2 + tan³(x).

If so, then d/dx(2 + tan³(x)) = d/dx(2) + d/dx(tan³(x)), using the sum rule.

Now use the chain rule to get d/dx(tan³(x)).

Why do you want to integrate? The power rule for integrals says that
\int u^n du~=~ \frac{u^{n + 1}}{n + 1}~for~n~\neq~-1

There's a different formula to use if n = -1.

Question for d/dx(tan³(x)) <- Y chain rule? There is only 1 term and a power why is it we can't use power rule?

Secondly, what's the different formula for n = -1?

Sorry if I been asking a lot of stupid questions but me and my pals have been thrown into taking this module without any prior foundation knowledge, we don't know Trigo, angles and everything. And the teachers at collage only touch on stuff briefly. I've been rewatching lecture videos over and over again and they still didn't explain anything. So I been searching the net for stuff. Sorry again if I asked too much.

 

[LumenPlacidum]

You have to use the chain rule for tan^3(x) because you have two functions there:
f(u) = u^3
g(x) = tan(x)

tan^3(x) is (tan(x))^3, which is f(g(x)).

When you take the derivative of tan^3(x), you have to take the derivative of u^3 with respect to u, which gives 3u^2, then multiply by the derivative of tan(x), which is sec^2(x). Then substitute back in for u.

I would recommend approaching your teacher about this stuff. If you've never done trigonometry and you're having trouble with function composition, perhaps you're taking the wrong class.

 

[DorumonSg]

You have to use the chain rule for tan^3(x) because you have two functions there:
f(u) = u^3
g(x) = tan(x)

tan^3(x) is (tan(x))^3, which is f(g(x)).

When you take the derivative of tan^3(x), you have to take the derivative of u^3 with respect to u, which gives 3u^2, then multiply by the derivative of tan(x), which is sec^2(x). Then substitute back in for u.

I would recommend approaching your teacher about this stuff. If you've never done trigonometry and you're having trouble with function composition, perhaps you're taking the wrong class.

So we dun view it as tan^3(x) like x^3? So u mean tan^3(x) is different from tanX^3?

So wads the formula for n=-1 for Integration, I can't seen find it. I have this du/u^2 but I am kinda stuck there.

Nah, its the education system over here. There's a split during our so called High School, where students get the choose to continue High School or go for vocational stuff. The high school usually prepares them for collage, however high scorers like 15% of the ppl in the vocational school are given a chance to enter too. However uni is pretty much catered to the high school ppl, so when it comes to maths most of us suffer.

 

[Mark44]

So we dun view it as tan^3(x) like x^3? So u mean tan^3(x) is different from tanX^3?

tan³(x) is different from tan(x³). The first is notation that really means (tan(x))³, or (tan(x)*tan(x)*tan(x)). The second means take the cube of x, then take the tangent of the result.

So wads the formula for n=-1 for Integration, I can't seen find it. I have this du/u^2 but I am kinda stuck there.

The two formulas relating to the power rule for integration are
\int u^n du~=~ \frac{u^{n + 1}}{n + 1} + C~for~n~\neq~-1

and
\int u^{-1} du~=~ln|u| + C

To integrate 1/u², rewrite this as u^-2 and use the first formula above.

 

[DorumonSg]

To integrate 1/u², rewrite this as u^-2 and use the first formula above.

Hey I saw that! But the correct answer I have is -(1/U) + C how did that happen? Where's the In and stuff?

The initial part was U/(U^2) then after integration the answer is (-1/U) + C, no In?

 

[Mark44]

That's Ln, not In. Ln stands for natural logarithm (in Latin, with the order reversed).
\int \frac{du}{u^2}~=~\int u^{-2} du~=~\frac{u^{-1}}{-1} + C~=~\frac{-1}{u} + C

It is NOT true that the integral of du over whatever is Ln|whatever|. It's important to understand the "fine print" in the power rule for integrals; when you can apply it and when you can't.