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Current time:0:00Total duration:3:42

in the last video we saw the quotient rule the quotient rule which once again I have mixed feelings about because it really comes straight out of product rule if we have something of the form f of X over G of X that the derivative of it could be this business right over here so I thought I would at least do one example where we can apply that and we could do is it do it to find the derivative of something useful so what's the derivative what's the derivative with respect to X let me write this a little bit neater the derivative with respect to X of tangent of X and you might say hey Sal Way guy I thought this was about the quotient rule but you just have to remember what is the definition of the tangent of X well or what is one way to view the tangent of X well tangent of X is the same thing as sine of X and let me run out a color code it is the same thing as sine of X sine of X sine of X over cosine of X over cosine cosine of X and now it makes it makes it it looks clear that our expression is the ratio or it's a kind of a it's it's one function over another function so now we can just apply the quotient rule so the all of this business is going to be equal to the derivative of sine of X times cosine of X so what's the derivative of sine of X well that's just cosine of X so it's cosine of X is derivative of sine of X times whatever function we had in the denominator so times cosine of X cosine of X minus - whatever function we had in the numerator sine of X sine of X sine of X times the derivative of whatever we have in the denominator well what's the derivative of cosine of X well the derivative of cosine of X is negative sine of X so I'll put the sine of X here and it's a negative so I can just make this right over here a positive and then all of that all of that over whatever was in the denominator squared all of that over cosine cosine of X cosine of x squared now what does this simplify to the numerator right over here we have cosine of X times cosine of X so all of this simplifies to cosine squared of X and sine of x times sine of X that's just sine squared of X and what's cosine squared of X plus sine squared of X this is one of the most basic trigonometric identities it comes straight out of the unit circle definition of trig functions cosine squared let me write it over here cosine squared of X plus sine squared of X is equal to 1 is equal to 1 which simplifies things quite nicely so cosine of square cosine squared of X plus sine squared of X all of this entire numerator is equal to 1 so this nicely simplifies to 1 over 1 over cosine of x squared which we could also write like this cosine squared of X these are two ways of writing cosine of x squared which is the same exact thing as 1 over cosine of x squared which is the same thing as secant one over cosine of X is just secant of X secant of x squared or we could write it like this secant squared of X and so that's where it comes from if you know that the derivative of sine of X is cosine of X and the derivative of cosine of X is negative sine of X we can use the quotient rule which is once again comes straight out of the product rule to find the derivative of tangent X is secant squared of X