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An older video where Sal finds the derivative of tan(x) using the quotient rule. Created by Sal Khan.
Video transcript
In the last video, we saw that the quotient rule, which, once again, I have mixed feelings about because it really comes straight out of the product rule. If we have something in the form f of x over g of x, then 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 can could do it to find the derivative of something useful. So 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, wait, I thought this was about the quotient rule. But you just have to remember, what is the definition of the tangent of x? Or what is one way to view the tangent of x? The tangent of x is the same thing as sine of x-- let me now color code it-- is the same thing as sine of x over cosine of x. And now it looks clear that our expression is the ratio or it's one function over another function. So now we can just apply the quotient rule. So 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 minus whatever function we had in the numerator, 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 we'll put the sine of x here. And it's a negative so I could just make this right over here a positive. And then all of that over, whatever was in the denominator squared, all of that over cosine of x squared. Now what does this simplify to? Well in 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. Let me write it over here. Cosine squared of x plus sine squared of x is equal to 1, which simplifies things quite nicely. So 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 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. 1 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, once again, comes straight out of the product rule to find the derivative of tangent x is secant squared of x.