Derivitive of Dot product

[steller]

let $\displaystyle u(t) =< -\sqrt{t}sint, t, t^{2/3}$ and $\displaystyle v(t) = < -\sqrt{t}sint, cos^2t, -t^{1/3}>$

Compute $\displaystyle {d/dt} (u(t) . v(t))$

Taking the dot product i have:

$\displaystyle tsint^2(t) + tcos^2(t) - t = 0$

From here i think i see a trig identity but not sure what to do with the t in front of cos, sin.

So, we will proceed with another route!

Remove whats common t . The negative 1 comes from -t from moving whats common.
$\displaystyle t(cos^2(t) - 1 + sin^2(t))=0$

So now i have

$\displaystyle t=0$ and $\displaystyle cos^2(t)-1 + sin^2(t) = 0$

wolfram said that this was an identity but i could not verify that!

$\displaystyle cos^2(t)-1 = 1-sin^2(t)$

plugging in the identity for cos^2(t) we have

$\displaystyle 1-sin^2(t)-1 + sin^2(t) = 0$

Therefore, $\displaystyle t=0$ and $\displaystyle 0=0$

making the derivative 0?

However, i never took the derivative and i couldn't verify the identity.

Right after i take the dot product could i move the -t to the right and divide out the other t?
then use the cos^2 + sin^2 = 1 identity? then take the derivative?

Where did i go wrong?

 

[HallsofIvy]

let $\displaystyle u(t) =< -\sqrt{t}sint, t, t^{2/3}$ and $\displaystyle v(t) = < -\sqrt{t}sint, cos^2t, -t^{1/3}>$

Compute $\displaystyle {d/dt} (u(t) . v(t))$

Taking the dot product i have:

$\displaystyle tsint^2(t) + tcos^2(t) - t = 0$

From here i think i see a trig identity but not sure what to do with the t in front of cos, sin.

Yes, you can factor out the "t" to get $\displaystyle t(sin^2(t)+ cos^2(t))- t= t- t= 0$ for all t.

So, we will proceed with another route!

Remove whats common t . The negative 1 comes from -t from moving whats common.
$\displaystyle t(cos^2(t) - 1 + sin^2(t))=0$

That is, as I said before, identically equal to 0, but since you say you did not know that, where did you get the "0" on the right?

So now i have

$\displaystyle t=0$ and $\displaystyle cos^2(t)-1 + sin^2(t) = 0$

wolfram said that this was an identity but i could not verify that!

$\displaystyle cos^2(t)-1 = 1-sin^2(t)$

plugging in the identity for cos^2(t) we have

$\displaystyle 1-sin^2(t)-1 + sin^2(t) = 0$

Therefore, $\displaystyle t=0$ and $\displaystyle 0=0$

making the derivative 0?

However, i never took the derivative and i couldn't verify the identity.

$\displaystyle sin^2(t)+ cos^2(t)= 1$!

Right after i take the dot product could i move the -t to the right and divide out the other t?
then use the cos^2 + sin^2 = 1 identity? then take the derivative?

Where did i go wrong?

Since that dot product is identically 0, its derivative is identically 0.

 

[steller]

Thank you i think i see it now.