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# Derivative of tan(x) (old)

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.