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# Derivatives of tan(x) and cot(x)

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.B (LO)
,
FUN‑3.B.3 (EK)

## Video transcript

we already know the derivatives of sine and cosine we know that the derivative with respect to X of sine of X is equal to cosine of X we know the derivative with respect to X of cosine of X is equal to negative sine of X and so what we want to do in this video is find the derivatives of the other basic trig functions so in particular we know let's figure out what the derivative with respect to X let's first do tangent of X tangent of X well this is the same thing as trying to find the derivative with respect to X of well tangent of X is just sine of X sine of X over cosine of X and since it can be expressed as a as as the quotient of two functions we can apply the quotient rule here to to evaluate this or to figure out what this is going to be the quotient rule tells us that this is going to be the derivative of the top function which we know is cosine of X times the bottom function which is cosine of X so times cosine of X minus minus the top function which is sine of X sine of X times the derivative of the bottom function so the derivative of cosine of X is negative sine of X so I could put the sine of X there but where the negative can just cancel that out and it's going to be over over the bottom function squared so cosine squared of X now what is this well what we have here this is just cosine squared of X this is just sine squared of X and we know from the Pythagorean identity and this is really just thought of comes out of the unit circle definition the cosine squared of X plus sine squared of X well that's going to be equal to 1 for any X so all of this is equal to 1 and so we end up with 1 over our cosine squared X which is the same thing as which is the same thing as secant of x squared 1 over cosine of X is secant so this is just secant of x squared so that was pretty straightforward now let's just do the inverse of the or you could say the reciprocal I should say of the tangent function which is the cotangent so that was fun so let's do that DDX of cotangent not cosign of cotangent of X well same idea that's the derivative with respect to X and this time so let me make some sufficiently large brackets so now this is cosine of X over sine of X over sine of X but once again we can use the quotient rule here so this is going to be the derivative of the top function which is negative we did in that magenta color that is negative sine of X times the bottom function so times sine of X sine of X minus minus the top function cosine of X cosine of X times the derivative of the bottom function which is just going to be another cosine of X and then all of that over the bottom function squared so the sine of x squared now what does this simplify to up here let's see this is sine squared of X although we have a negative there minus cosine squared of X but we could factor out the negative and this would be negative sine squared of X plus cosine squared of X well this is just 1 by the Pythagorean identity and so this is negative 1 over sine squared X negative 1 over sine squared X and that is the same same thing as negative cosecant squared of I'm running out of space of of X there you go
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