- We alreadyknow the derivatives of sine & cosine. We know that the derivativewith respect to x of sine of x is equal to cosine of x. We know that the derivativewith respect lớn x of cosine of x is equalto negative sine of x. And so what we want to vị in this clip is find the derivatives of theother basic trig functions. So, in particular, weknow, let's figure out what the derivative with respect khổng lồ x, let's first bởi vì tangent of x. Tangent of x, well this is the same thing as trying lớn find thederivative with respect lớn x of, well, tangent of x is just sine of x, sine of x over cosine of x. Và since it can be expressed as the quotient of two functions, we can apply the quotientrule here to lớn evaluate this, or lớn figure out what this is going khổng lồ be. The quotient rule tells usthat this is going to be the derivative of the top function, which we know is cosine ofx times the bottom function which is cosine of x, so times cosine of x minus, minus the topfunction, which is sine of x, sine of x, times the derivativeof the bottom function. So the derivative of cosineof x is negative sine of x, so I can put the sine of x there, but where the negativecan just cancel that out. And it's going lớn be over, over the bottom function squared. So cosine squared of x. Now, what is this? Well, what we have here, thisis just a cosine squared of x, this is just sine squared of x. & we know from the Pythagorean identity, và this is really just out of, comes out of the unit circle definition, the cosine squared of xplus sine squared of x, well that's gonna beequal khổng lồ one for any x. So all of this is equal lớn one. & so we end up with oneover cosine squared x, which is the same thing as,which is the same thing as, the secant of x squared. One over cosine of x is secant, so this is just secant of x squared. So that was pretty straightforward. Now, let's just vày the inverse of the, or you could say thereciprocal, I should say, of the tangent function,which is the cotangent. Oh, that was fun, so let's bởi that, d dx of cotangent, not cosine, of cotangent of x. Well, same idea, that's thederivative with respect khổng lồ x, and this time, let me make somesufficiently large brackets. So now this is cosine of x over sine of x, over sine of x. But once again, we can usethe quotient rule here, so this is going khổng lồ be thederivative of the đứng đầu function which is negative, use that magenta color. That is negative sine of x times the bottom function, so times sine of x, sine of x, minus, minus the vị trí cao nhất function, cosine of x, cosine of x, times thederivative of the bottom function which is just going tobe another cosine of x, and then all of that overthe bottom function squared. So sine of x squared. Now what does this simplify to? Up here, let's see, thisis sine squared of x, we have a negative there, minus cosine squared of x. But we could factor out the negative và this would benegative sine squared of x plus cosine squared of x. Well, this is just one bythe Pythagorean identity, và so this is negativeone over sine squared x, negative one over sine squared x.


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Và that is the same thing as negative cosecant squared, I'm running out of space, of x. There you go.