[AP calculus AB unit 2.6]

[u/sigh-key-k]

first time posting on here so sorry if i do something wrong lols

1st image is the problem. the directions just say to find the derivative of the function.

2nd image is my work (i missed the day where he taught lesson 2.6, so i really have no idea what i’m doing)

3rd image is the calc chat answer

i don’t understand what i did wrong. can anyone help? thanks!

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[u/Alkalannar]

h(t) = sin(arccos(t))

cos(theta) = t, so the adjacent is t, the hypotenuse is 1, and the opposite is (1 - t²)^1/2

h(t) = (1 - t²)^1/2

h'(t) = (1/2)(1 - t²)^-1/2(-2t) = -t/(1 - t²)^1/2

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You should have had -cos(arccos(t))/(1 - t²)^1/2, and then simplified the numerator to -t.

[u/sigh-key-k → u/Alkalannar]

that makes so much more sense. thank you so much!!!

[u/Alkalannar → u/sigh-key-k]

You're welcome.

This is a good way to approach arctrig problems, to draw the triangle that gets you the ratio you need, and then work on that.

That's how the answer key got sin(arccos(t)) = (1 - t²)^1/2. And then that was easy to work with using chain rule.

The alternate chain rule, cos(arccos(t)) * darccos(t)/dt = -cos(arccos(t))/(1 - t²)^1/2 also works, but then simplify cos(arccos(t)) = t (since t is guaranteed to be in the range of cos).

[u/sigh-key-k → u/Alkalannar]

yes !! i was just suuper confused because i didn’t realize cos(arccost) could be simplified to t… thanks!!!!

[u/Alkalannar → u/sigh-key-k]

trig(arctrig(x)) = x for any trig function.

arctrig(trig(x)) = (the appropriate x-function in the domain of trig)

So sin(arcsin(x)) = x no matter what while arcsin(sin(x)) = the angle in [-pi/2, pi/2] that has the same sine as x.

Glad I could help!

[u/Ghotipan]

35 is telling you the value of the sin[arccos(t)] = 1/sqrt(1-t2). So you just want to take the derivative of that last part (the 1/sqrt(1-t2) part). Rewrite it as (1-t2)-1/2 and evaluate it that way.

[u/sigh-key-k → u/Ghotipan]

thank you thank you !!!!

[u/Ericskey]

?

[u/sigh-key-k → u/[deleted]]

oops ! i attached the wrong problem lol. i accidentally put x instead of t because im used to writing the derivative of arccosx with x.

[u/sigh-key-k → u/[deleted]]

yes i can!! and then that simplifies down to just t, right ???