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# Derivatives of sec(x) and csc(x)

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

## Video transcript

in a previous video we use the quotient rule in order to find the derivatives of tangent of X and cotangent of X and what I want to do in this video is to keep going and find the derivatives of secant of X and cosecant of X so let's start with secant of X the derivative with respect to X of secant of X well secant of X is the same thing as so we're going to find the derivative with respect to X of secant of X is the same thing as 1 over 1 over the cosine of X and that's just the definition of secant and you could there's multiple ways you could do this when you learn the chain rule that actually might be a more natural thing to use to evaluate the derivative here but we know the quotient rule so we will apply the quotient rule here and it's no coincidence that you to the same answer the quotient rule actually can be derived based on the chain rule and the product rule but I won't keep going into that let's just apply the quotient rule right over here so this derivative is going to be equal to it's going to be equal to the derivative of the top well what's the derivative of 1 with respect to X well that's just 0 times the function on the bottom so times cosine of X cosine of X minus minus the function on the top well that's just 1 times the derivative on the bottom well the derivative of the bottom is derivative of cosine of X is negative sine of X so we could put the sine of X there but it's negative sine of X so that you have a minus and it would be a negative so we could just make that a positive and then all of that over the function on the bottom squared so cosine of x squared and so 0 times cosine of X that is just 0 and so all we are left with is sine of X over cosine of x squared and there's multiple ways that you could rewrite this if you like you could say that this is the same thing as sine of x over cosine of x times 1 over cosine X and of course this is tangent of x times secant of X secant of X so you could say derivative of secant X is sine of X over cosine squared of X or it is tangent of x times secant of X so now let's do cosecant so the derivative with respect to X of cosecant of X well that's the same thing as the derivative with respect to X of 1 over sine of X cosecant is 1 over sine of X I remember that because you think it's cosecant maybe it's it's the reciprocal of cosine but it's not it's the opposite of what you would expect cosines reciprocal isn't cosecant it is secant once again opposite of what you were out of what you would expect that starts with an S this starts with the C that starts with the C that starts with an S just the way that it happened to be defined but anyway let's just evaluate this once again we'll do the quotient rule but you could also do this using the chain rule so it's going to be the derivative of the expression on top which is zero times the expression on the bottom which is sine of X sine of X minus the expression on top which is just 1 times the derivative of the expression on the bottom which is cosine of X all of that over the expression on the bottom squared sine squared of X that's 0 so we get negative cosine of X over sine over sine squared of X so that's one way to think about it or if you like you could view this the same thing we did over here this is the same thing as negative cosine of X or sine of X times 1 over sine of X and this is negative cotangent of X negative cotangent of X times let me write this way times 1 over sine of X is cosecant of X cosecant cosecant of X so whichever one you find more useful