Thread: Pleae help me find these derivatives!

1. Pleae help me find these derivatives!

1. f(theta)= sec (theta) ^2
The answer is 2 theta sec^2 tan ^2 theta.

But I got 2sec ^2 theta tan theta.

2. y= sin^2 x- cos^2 x

But I got 4 sinxcosx.

2. Originally Posted by Power of One
1. f(theta)= sec (theta) ^2
The answer is 2 theta sec^2 tan ^2 theta.

But I got 2sec ^2 theta tan theta.

2. y= sin^2 x- cos^2 x

But I got 4 sinxcosx.

For the first one, is it $\sec\!\left(\theta^2\right)$ or $\sec^2\!\left(\theta\right)$?

Your second answer is correct, since $4\sin x\cos x=2(2\sin x\cos x)=2\sin\!\left(2x\right)$

3. Originally Posted by Chris L T521
For the first one, is it $\sec\!\left(\theta^2\right)$ or $\sec^2\!\left(\theta\right)$?

Your second answer is correct, since $4\sin x\cos x=2(2\sin x\cos x)=2\sin\!\left(2x\right)$
The first one is $\sec\!\left(\theta^2\right)$.

Can you explain to me how you got $4\sin x\cos x=2(2\sin x\cos x)=2\sin\!\left(2x\right)$?

4. Originally Posted by Power of One

But I got 4 sinxcosx.
$4\sin(x)\cos(x) = 2\times 2 \sin(x)\cos(x) = 2\times \sin(2x)$

5. Originally Posted by Power of One
The first one is $\sec\!\left(\theta^2\right)$.

Can you explain to me how you got $4\sin x\cos x=2(2\sin x\cos x)=2\sin\!\left(2x\right)$?
If that is the case, then by chain rule, we get

$f^{\prime}\!\left(\theta\right)\sec\!\left(\theta^ 2\right)\tan\!\left(\theta^2\right)\cdot2\theta=2\ theta\sec\!\left(\theta^2\right)\tan\!\left(\theta ^2\right)$.

To answer your second question, its a double angle identity: $\sin\!\left(2\theta\right)=2\sin\theta\cos\theta$.

From you answer, it followed that $4\sin\theta\cos\theta=2\left(2\sin\theta\cos\theta \right)=2\sin\!\left(2\theta\right)$.

Does this clarify things?

6. Originally Posted by Power of One

Can you explain to me how you got $4\sin x\cos x=2(2\sin x\cos x)=2\sin\!\left(2x\right)$?
It is a known trig identity. You should commit it to memory.

7. Thank you guys for reminding me to memorize my trig. identities and double-angle formulas!

@Chris L T521- I still don't understand the first problem. Using the chain rule,

I would bring 2 to the front minus one a power from sec theta ^2 to be left with sec theta, then multiply by its derivative sec theta tan theta.

What am I doing wrong here?

8. Originally Posted by Power of One
Thank you guys for reminding me to memorize my trig. identities and double-angle formulas!

@Chris L T521- I still don't understand the first problem. Using the chain rule,

I would bring 2 to the front minus one a power from sec theta ^2 to be left with sec theta, then multiply by its derivative sec theta tan theta.

What am I doing wrong here?
That's a pitfall you should stay away from: $\sec^2\theta\neq\sec\!\left(\theta^2\right)$

To make it a little easier, let $u=\theta^2$.

So when we differentiate $\sec u$ with respect to theta, we have:

$f^{\prime}\!\left(\theta\right)=\sec u\tan u\cdot\frac{\,du}{\,d\theta}$

Since $u=\theta^2,\,\frac{\,du}{\,d\theta}=2\theta$.

So, $f^{\prime}\!\left(\theta\right)=\sec\!\left(\theta ^2\right)\tan\!\left(\theta^2\right)\cdot 2\theta=2\theta\sec\!\left(\theta^2\right)\tan\!\l eft(\theta^2\right)$.

Does this clarify things?

9. Originally Posted by Chris L T521
That's a pitfall you should stay away from: $\sec^2\theta\neq\sec\!\left(\theta^2\right)$

To make it a little easier, let $u=\theta^2$.

So when we differentiate $\sec u$ with respect to theta, we have:

$f^{\prime}\!\left(\theta\right)=\sec u\tan u\cdot\frac{\,du}{\,d\theta}$

Since $u=\theta^2,\,\frac{\,du}{\,d\theta}=2\theta$.

So, $f^{\prime}\!\left(\theta\right)=\sec\!\left(\theta ^2\right)\tan\!\left(\theta^2\right)\cdot 2\theta=2\theta\sec\!\left(\theta^2\right)\tan\!\l eft(\theta^2\right)$.

Does this clarify things?
I'm sorry, but no it doesn't. I tried to follow you, but I got lost. Where did du/ d theta come from?

10. Originally Posted by Power of One
I'm sorry, but no it doesn't. I tried to follow you, but I got lost. Where did du/ d theta come from?
Its a result from the chain rule:

$\frac{\,df}{\,d\theta}=\frac{\,df}{\,du}\frac{\,du }{\,d\theta}$

where, in our problem, $f=\sec u$ and $u=\theta^2$

11. I FINALLY UNDERSTAND IT! Thank you so much for being so patient with my stupid self. I really appreciate the help!