yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Trig functions differentiation | Derivative rules | AP Calculus AB | Khan Academy


4m read
·Nov 11, 2024

So let's say that we have ( y ) is equal to the secant of (\frac{3\pi}{2} - x), and what we want to do is we want to figure out what (\frac{dy}{dx}) is, the derivative of ( y ) with respect to ( x ) at ( x = \frac{\pi}{4} ).

Like always, pause this video and see if you could figure it out. Well, as you can see here, we have a composite function; we're taking the secant not just of ( x ), but you could view this as of another expression that I guess you could define or as of another function.

So, for example, if we call this right over here ( u(x) ), so let's do that. If we say ( u(x) ) is equal to (\frac{3\pi}{2} - x), we could also figure out ( u' ) of ( x ) is going to be equal to the derivative of (\frac{3\pi}{2}); that's just going to be zero. The derivative of (-x) is going to be (-1), and you could just view that as a power rule; it's ( 1 \cdot -1 \cdot x^{0} ), which is just one.

So there you go! We could view this as the derivative of secant with respect to ( u(x) ), and when we take the derivative, the derivative of secant with respect to ( u(x) ) times the derivative of ( u ) with respect to ( x ).

You might say, "Well, what about the derivative of secant?" Well, in other videos, we actually prove it out, and you could actually re-derive it. Secant is just ( \frac{1}{\cos(x)} ), so it comes straight out of the chain rule.

So in other videos, we proved that the derivative of the secant of ( x ) is equal to (\sec(x) \tan(x)). So if we're trying to find the derivative of ( y ) with respect to ( x ), well, it's going to be the derivative with respect to ( u(x) ) times the derivative of ( u ) with respect to ( x ).

So let's do that. The derivative of secant with respect to ( u(x) ) well, instead of seeing an ( x ) everywhere, you're going to see a ( u(x) ) everywhere. So this is going to be (\sec(u(x)) \tan(u(x))).

I don't have to write ( u(x) ); I could write (\frac{3\pi}{2} - x), but I'll write ( u(x) ) right over here just to really visualize what we're doing: (\sec(u(x)) \tan(u(x))).

So that's the derivative of secant with respect to ( u(x) ), and then the chain rule tells us it's going to be that times ( u' ). ( u' ) of ( x ) we already figured out is (-1), so I could write (\sec(u(x)) \tan(u(x)) \cdot u' ) where ( u' ) of ( x ) we already figured out is (-1).

Now, we want to evaluate ( \frac{dy}{dx} ) at ( x = \frac{\pi}{4} ). So when that is equal to ( \frac{pi}{4} ), let's see. This is going to be equal to (\sec\left(\frac{3\pi}{2} - \frac{\pi}{4}\right)\tan\left(\frac{3\pi}{2} - \frac{\pi}{4}\right) \cdot -1).

So if you have a common denominator, that is (\frac{6\pi}{4} - \frac{\pi}{4} = \frac{5\pi}{4}). So it's (\sec\left(\frac{5\pi}{4}\right) \tan\left(\frac{5\pi}{4}\right) \cdot -1).

Now, what is (\sec\left(\frac{5\pi}{4}\right)) and (\tan\left(\frac{5\pi}{4}\right))? Well, I don't have that memorized, but let's actually draw a unit circle, and we should be able to figure out what that is.

So a unit circle... I try to hand-draw it as best as I can; please forgive me that this circle does not look really like a circle! Alright, okay, so let me just remember my angles. In my brain, I sometimes convert into degrees. (\frac{\pi}{4}) is (45°), this is (\frac{\pi}{2}), this is (\frac{3\pi}{4}), this is (\frac{4\pi}{4}), this is (\frac{5\pi}{4}), lands you right over there.

So if you wanted to see where this intersects the unit circle, this is at the point where your ( x )-coordinate is (-\frac{\sqrt{2}}{2}) and your ( y )-coordinate is (-\frac{\sqrt{2}}{2}).

If you're wondering how I got that, I encourage you to review the unit circle and some of the standard angles around the unit circle; you'll see that in the trigonometry section of Khan Academy. But this is enough for us because the sine is the ( y )-coordinate. So (\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}).

So this is (-\frac{\sqrt{2}}{2}), and then the cosine is the ( x )-coordinate, which is also (-\frac{\sqrt{2}}{2}), but it's going to be that squared: (\left(-\frac{\sqrt{2}}{2}\right)).

So if we square this, it's going to become positive, and then (\left(-\frac{\sqrt{2}}{2}\right)^{2} = \frac{2}{4} = \frac{1}{2}), so this is the denominator.

In the numerator, the negative cancels out with that negative, and so we are left with—and we deserve a little bit of a drum roll— that we are left with (\frac{-\frac{\sqrt{2}}{2}}{\frac{1}{2}}).

Well, that's the same thing as multiplying by (2), so we are left with (\sqrt{2}). This is the slope of the tangent line to the graph of ( y ) is equal to this when ( x ) is equal to (\frac{\pi}{4}). Pretty exciting!

More Articles

View All
Constructing hypotheses for a significance test about a proportion | AP Statistics | Khan Academy
We’re told that Amanda read a report saying that 49% of teachers in the United States were members of a labor union. She wants to test whether this holds true for teachers in her state, so she is going to take a random sample of these teachers and see wha…
Badland's Prairie Dogs vs Coyote | America's National Parks | National Geographic
NARRATOR: Badlands National Park, South Dakota, 244,000 acres split into two dramatic worlds, the Rocky Badlands themselves, carved out of the ground by wind and rain, and beyond them, an ancient sea of grass, home to the icons of the Old West. This land …
how I got rid of my ACNE after 8 years - ONLY thing worked
If drinking more water, exercising more, sleeping more, reducing stress, or generally the tips that people give you on YouTube didn’t work for you then this video is for you. Hi, guys, it’s me Ruri. Today, I’m back with another very requested video. Just …
Assassination politics: Not inevitable
In my previous video, I described Jim Bell’s idea of assassination politics and said that I agreed with him that the emergence of such a system seemed inevitable. Thanks to the user, peace requires anarchy. I’ve since read an article by Bob Murphy, which …
How to Poop on a Nuclear Submarine - Smarter Every Day 256
Hey, it’s me, Destin. Welcome back to Smarter Every Day. If you’re just joining this nuclear submarine deep dive series, boy have I got a treat for you. We have covered a ton of stuff that happens on board nuclear submarines. We looked at sonar, we looked…
Kevin O'Leary's Predictions for 2022: Are we ready for what's coming next year?
[Music] He is the chairman of O’Leary Financial Group. He is a Shark Tank investor. He is a friend of the show. Mr. Wonderful is back to give us his, uh, I guess wrap up on what has been a pretty impressive year to say the least. Kevin will have, uh, you …